To understand the role of electronic nematic order in the interplay between s- and d-wave particle-particle or particle-hole condensate states, relations between various s- and d-wave order parameters are studied. We find that the nematic operator transforms two independent six-dimensional vectors. The d-wave superconducting, d-density wave, and antiferromagnetic orders are organized into one vector, and the s-wave superconducting, charge density wave, and spin-triplet d-density wave orders into the other vector. Each vector acts as a superspin and transforms under the action of SO(6) where charge, spin, eta- and pi-pairing, spin-triplet nematic operators satisfy the SO(6) Lie algebra. Electronic nematic order is not a part of the SO(6) group. It commutes with all 15 generators. Our findings imply that nematic order does not affect the competition among the order parameters within the same superspin, while it strongly interferes the interplay between two order parameters that belong to different superspins. For example, nematicity allows a linear coupling between d- and s-wave superconducting order parameters which modifies the superconducting transition temperature. A generalized Ginzburg-Landau theory and further physical implications are discussed.
Strong interplay of spin and charge/orbital degrees of freedom is the fundamental characteristic of the iron-based superconductors (FeSCs), which leads to the emergence of a nematic state as a rule in the vicinity of the antiferromagnetic state. Despite intense debate for many years, however, whether nematicity is driven by spin or orbital fluctuations remains unsettled. Here, by use of transport, magnetization, and $^{75}$As nuclear magnetic resonance (NMR) measurements, we show a striking transformation of the relationship between nematicity and spin fluctuations (SFs) in Na$_{1-x}$Li$_x$FeA…
We present $^{77}$Se-NMR measurements on single-crystalline FeSe under pressures up to 2 GPa. Based on the observation of the splitting and broadening of the NMR spectrum due to structural twin domains, we discovered that static, local nematic ordering exists well above the bulk nematic ordering temperature, $T_{\rm s}$. The static, local nematic order and the low-energy stripe-type antiferromagnetic spin fluctuations, as revealed by NMR spin-lattice relaxation rate measurements, are both insensitive to pressure application. These NMR results provide clear evidence for the microscopic cooperat…
Motivated by the recent experiments suggesting the importance of nematicity in the phase diagrams of ironbased and cuprate high-Tc superconductors, we study the influence of nematicity on the collective modes inside the superconducting state in a non-equilibrium. In particular, we consider the signatures of collective modes in short-time dynamics of a system with competing nematic and s- and d-wave superconducting orders. In the rotationally symmetric state, we show that the Bardasis-Schrieffer mode, corresponding to the subdominant pairing, hybridizes with the nematic collective mode and merg…
Nematicity and magnetism are two key features in Fe-based superconductors, and their interplay is one of the most important unsolved problems. In FeSe, the magnetic order is absent below the structural transition temperature $T_{str}=90$K, in stark contrast that the magnetism emerges slightly below $T_{str}$ in other families. To understand such amazing material dependence, we investigate the spin-fluctuation-mediated orbital order ($n_{xz}\neq n_{yz}$) by focusing on the orbital-spin interplay driven by the strong-coupling effect, called the vertex correction. This orbital-spin interplay is v…
Nematic order has manifested itself in a variety of materials in the cuprate family. We propose an effective field theory of a layered system with incommensurate, intertwined spin- and charge-density wave (SDW and CDW) orders, each of which consists of two components related by $C_4$ rotations. Using a variational method (which is exact in a large $N$ limit), we study the development of nematicity from partially melting those density waves by either increasing temperature or adding quenched disorder. As temperature decreases we first find a transition to a nematic phase, but depending on the r…
Elucidating the microscopic origin of nematic order in iron-based superconducting materials is important because the interactions that drive nematic order may also mediate the Cooper pairing. Nematic order breaks fourfold rotational symmetry in the iron plane, which is believed to be driven by either orbital or spin degrees of freedom. However, as the nematic phase often develops at a temperature just above or coincides with a stripe magnetic phase transition, experimentally determining the dominant driving force of nematic order is difficult. Here, we use neutron scattering to study structura…
We report high-resolution measurements of the in-plane thermal expansion anisotropy in the vicinity of the electronic nematic phase in Sr$_3$Ru$_2$O$_7$ down to very low temperatures and in varying magnetic field orientation. For fields applied along the c-direction, a clear second-order phase transition is found at the nematic phase, with critical behavior compatible with the two-dimensional Ising universality class (although this is not fully conclusive). Measurements in a slightly tilted magnetic field reveal a broken four-fold in-plane rotational symmetry, not only within the nematic phase…
Symmetry breaking is a fundamental concept in condensed matter physics whose presence often heralds new phases of matter. For instance, the breaking of time reversal symmetry is traditionally linked to magnetic phases in a material, while the breaking of gauge symmetry can lead to superfluidity/superconductivity. Nematic phases are phases in which rotational symmetry is broken while maintaining translational symme- try, and are traditionally associated with liquid crystals. Electronic nematic states where the or- thogonal in-plane crystal directions have different electronic properties have ga…
In common with other iron-based high temperature superconductors, FeSe exhibits a transition to a ``nematic'' phase below 90Kelvin in which the crystal rotation symmetry is spontaneously broken. However, the absence of strong low-frequency magnetic fluctuations near or above the transition has been interpreted as implying the primacy of orbital ordering. In contrast, we establish that quantum fluctuations of spin-1 local moments with strongly frustrated exchange interactions can lead to a nematic quantum paramagnetic phase consistent with the observations in FeSe. We show that this phase is a …
We study the in-plane anisotropy of the thermoelectric power and electrical resistivity on detwinned single crystals of isovalent substituted EuFe$_{2}$(As$_{1-x}$P$_{x}$)$_2$. Compared to the resistivity anisotropy the thermopower anisotropy is more pronounced and clearly visible already at temperatures much above the structural and magnetic phase transitions. Most remarkably, the thermopower anisotropy changes sign below the structural transition. This is associated with the interplay of two contributions due to anisotropic scattering and orbital polarization, which dominate at high- and low…
The electronic nematic order characterized by broken rotational symmetry has been suggested to play an important role in the phase diagram of the high temperature cuprates. We study the interplay between the electronic nematic order and a spin density wave order in the presence of a magnetic field. We show that a cooperation of the nematicity and the magnetic field induces a finite coupling between the spin density wave and spin-triplet staggered flux orders. As a consequence of such a coupling, the magnon gap decreases as the magnetic field increases, and it eventually condenses beyond a crit…
Strongly interacting electrons in solid-state systems often display tendency towards multiple broken symmetries in the ground state. The complex interplay between different order parameters can give rise to a rich phase diagram. Here, 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 crossing the superconducting dome, a reduction of the critical tempera…
Recent experiments have explored two-dimensional electron gases (2DEGs) at oxide (111) surfaces and interfaces, finding evidence for hexagonal symmetry breaking in SrTiO$_3$ at low temperature. We discuss many-body instabilities of such (111) 2DEGs, incorporating multiorbital interactions in the $t_{2g}$ manifold which can induce diverse magnetic and orbital orders. Such broken symmetries may partly account for the observed nematicity, cooperating or competing with phonon mechanisms. We present an effective field theory for the interplay of magnetism and nematic charge order, and discuss impli…
Fermi surface fluctuations and lattice instabilities in the 2D metallic kagome superconductor CsV$_3$Sb$_5$ are elucidated via polarization-resolved Raman spectroscopy. The presence of a weak electronic continuum in high-quality samples marks the cross-over into the charge-density-wave (CDW) ordered phase, while impurity-rich samples promote strong defect-induced electronic scattering processes that affect the coherence of the CDW phase. CDW-induced phonon anomalies appear below $T_{\mathrm{CDW}}$, with emergent $C2$ symmetry for one of the CDW amplitude modes, alluding to nematicity. In conju…
Evidence of nematic effects in the mixed superconducting phase of slightly underdoped $\text{Ba}(\text{Fe}_{1-x}\text{Co}_x)_2\text{As}_2$ is reported. We have found strong in-plane resistivity anisotropy for crystals in different strain conditions. For these compositions, there is no magnetic long range order, so the description may be ascribed to the interplay between the superconducting and nematic order parameters. A piezoelectric-based apparatus is used to apply tensile or compressive strain to tune nematic domain orientation in order to examine intrinsic nematicity. Measurements are done…
To understand the rich electronic phase diagram in FeSe under pressure that vividly demonstrates the strong interplay between the nematicity, magnetism and superconductivity, we analyze the electronic states by including the higher-order many-body effects called the vertex correction (VC). We predict the pressure-induced emergence of xy-orbital hole-pocket based on the first-principles analysis. Due to this pressure-induced Lifshitz transition, the spin fluctuations on the xy orbital are enhanced, whereas those on xz,yz orbitals are gradually reduced. For this reason, nonmagnetic orbital order…
Motivated by the recent angle-resolved photoemission spectroscopy (ARPES) on FeSe and iron pnictide families of iron-based superconductors, we have studied the orbital nematic order and its interplay with antiferromagnetism within the two-orbital Hubbard model. We used random phase approximation (RPA) to calculate the dependence of the orbital and magnetic susceptibilities on the strength of interactions and electron density (doping). To account for strong electron correlations not captured by RPA, we further employed non-perturbative variational cluster approximation (VCA) capable of capturin…
In the high-transition-temperature (high-Tc) superconductors the pseudogap phase becomes predominant when the density of doped holes is reduced1. Within this phase it has been unclear which electronic symmetries (if any) are broken, what the identity of any associated order parameter might be, and which microscopic electronic degrees of freedom are active. Here we report the determination of a quantitative order parameter representing intra-unit-cell nematicity: the breaking of rotational symmetry by the electronic structure within CuO2 unit cell. We analyze spectroscopic-imaging scanning tunn…
The pseudogap phenomenon in cuprates is the most mysterious puzzle in the research of high-temperature superconductivity. In particular, whether the pseudogap is associated with a crossover or phase transition has been a long-standing controversial issue. The tetragonal cuprate HgBa$_2$CuO$_{4+δ}$, with only one CuO$_2$ layer per primitive cell, is an ideal system to tackle this puzzle. Here, we measure the anisotropy of magnetic susceptibility within the CuO$_2$ plane with exceptionally high-precision magnetic torque experiments. Our key finding is that a distinct two-fold in-plane anisotropy…
An SDW antiferromagnetic (SDW-AF) low temperature phase transition is generally observe and the AF spin fluctuations are considered to play an important role for the superconductivity paring mechanism in FeAs superconductors. However, a similar magnetic phase transition is not observed in FeSe superconductors, which has caused considerable discussion. We report on the intrinsic electronic states of FeSe as elucidated by transport measurements under magnetic fields using a high quality single crystal. A mobility spectrum analysis, an ab initio method that does not make assumptions on the transp…
The interplay of orbital and spin degrees of freedom is the fundamental characteristic in numerous condensed matter phenomena, including high temperature superconductivity, quantum spin liquids, and topological semimetals. In iron-based superconductors (FeSCs), this causes superconductivity to emerge in the vicinity of two other instabilities: nematic and magnetic. Unveiling the mutual relationship among nematic order, spin fluctuations, and superconductivity has been a major challenge for research in FeSCs, but it is still controversial. Here, by carrying out 77Se nuclear magnetic resonance (…
The development of collective long-range order via phase transitions occurs by the spontaneous breaking of fundamental symmetries. Magnetism is a consequence of broken time-reversal symmetry while superfluidity results from broken gauge invariance. The broken symmetry that develops below 17.5K in the heavy fermion compound URu2Si2 has long eluded such identification. Here we show that the recent observation of Ising quasiparticles in URu2Si2 results from a spinor order parameter that breaks double time-reversal symmetry, mixing states of integer and half-integer spin. Such "hastatic order" hyb…
Layered kagome-lattice materials with 3$d$ transition metals provide a fertile playground for studies on geometry frustration, band topology and other novel ordered states. A representative class of materials AV$_3$Sb$_5$ (A=K, Rb, Cs) have been proved to possess various unconventional phases such as superconductivity, non-trivial $\mathbb{Z}_2$ band topology, and electronic nematicity, which are intertwined with multiple interlaced charge density waves (CDW). However, the interplay among these novel states and their mechanisms are still elusive. Recently, the discovery of isostructural titani…
Understanding the organizing principles of interacting electrons and the emergence of novel electronic phases is a central endeavor of condensed matter physics. Electronic nematicity, in which the discrete rotational symmetry in the electron fluid is broken while the translational one remains unaffected, is a prominent example of such a phase. It has proven ubiquitous in correlated electron systems, and is of prime importance to understand Fe-based superconductors. Here, we find that fluctuations of such broken symmetry are exceptionally strong over an extended temperature range above phase tr…
The electronic structure of the enigmatic iron-based superconductor FeSe has puzzled researchers since spectroscopic probes failed to observe the expected electron pocket at the $Y$ point in the 1-Fe Brillouin zone. It has been speculated that this pocket, essential for an understanding of the superconducting state, is either absent or incoherent. Here, we perform a theoretical study of the preferred nematic order originating from nearest-neighbor Coulomb interactions in an electronic model relevant for FeSe. We find that at low temperatures the dominating nematic components are of inter-orbit…
Iron-based superconductors are well-known for their intriguing phase diagrams, which manifest a complex interplay of electronic, magnetic and structural degrees of freedom. Among the phase transitions observed are superconducting, magnetic, and several types of structural transitions, including a tetragonal-to-orthorhombic and a collapsed-tetragonal transition. In particular, the widely-observed tetragonal-to-orthorhombic transition is believed to be a result of an electronic order that is coupled to the crystalline lattice and is, thus, referred to as nematic transition. Nematicity is therefo…
Electronic nematicity, the breaking of the crystal lattice rotational symmetry by the electronic fluid, is a fascinating quantum state of matter. Recently, BaNi$_2$As$_2$ has emerged as a promising candidate for a novel type of nematicity triggered by charge fluctuations. In this work, we scrutinize the electronic nematicity of BaNi$_2$(As$_{1-x}$P$_x$)$_2$ with $0 \leq x \leq 0.10$ using electronic transport measurements under strain. We report a large $B_{1g}$ elastoresistance coefficient that is maximized at a temperature slightly higher than the first-order triclinic transition, and that c…
As in between liquid and crystal phases lies a nematic liquid crystal, which breaks rotation with preservation of translation symmetry, there is a nematic superfluid phase bridging a superfluid and a supersolid. The nematic order also emerges in interacting electrons and has been found to largely intertwine with multi-orbital correlation in high-temperature superconductivity, where Ising nematicity arises from a four-fold rotation symmetry $C_4$ broken down to $C_2$. Here we report an observation of a three-state ($\mathbb{Z}_3$) quantum nematic order, dubbed "Potts-nematicity", in a system of…
In the magic angle twisted bilayer graphene (MATBG), one of the most remarkable observations is the $C_3$-symmetry-breaking nematic state. We identify that the nematicity in MATBG is the $E$-symmetry ferro bond order, which is the modulation of correlated hopping integrals owing to the $E$-symmetry particle-hole pairing condensation. The nematicity in MATBG originates from prominent quantum interference among $SU(4)$ valley+spin composite fluctuations. This novel "valley + spin fluctuation interference mechanism" is revealed by the density wave equation analysis for realistic multiorbital Hubb…
Tunable chiral and nematic states in the triple-Q antiferromagnet Co$_{1/3}$TaS$_2$
National High Magnetic Field Laboratory, Los Alamos National Lab（国家高磁场实验室，洛斯阿拉莫斯国家实验室）；Materials Science and Technology Division, Oak Ridge National Laboratory（橡树岭国家实验室材料科学与技术部）；Department of Physics and Astronomy, Seoul National University（首尔大学物理与天文学系）；Department of Physics and Astronomy, University of Tennessee, Knoxville（田纳西大学诺克斯维尔分校物理与天文学系）；Shull Wollan Center—A Joint Institute for Neutron Sciences, Oak Ridge National Laboratory（橡树岭国家实验室舒尔-沃兰中子科学联合研究所）；Institute of Applied Physics, Seoul National University（首尔大学应用物理研究所）
The superconductivity discovered in iron-pnictides is intimately related to a nematic ground state, where the C4 rotational symmetry is broken via the structural and magnetic transitions. We here study the nematicity in NaFeAs with the polarization dependent angle-resolved photoemission spectroscopy. A uniaxial strain was applied on the sample to overcome the twinning effect in the low temperature C2-symmetric state, and obtain a much simpler electronic structure than that of a twinned sample. We found the electronic structure undergoes an orbital-dependent reconstruction in the nematic state,…
We study the interplay of nematic and superconducting order in the two-dimensional Hubbard model and show that they can coexist, especially when superconductivity is not the energetically dominant phase. Due to a breaking of the $C_4$ symmetry, the coexisting phase inherently contains admixture of the $s$-wave pairing components. As a result, the superconducting gap exhibits very non-standard features including changed nodal directions. Our results also show that in the optimally doped regime the superconducting phase is typically unstable towards developing nematicity (breaking of the $C_4$ s…
The dominance of Coulomb interactions over kinetic energy of electrons in narrow, non-trivial moiré bands of magic-angle twisted bilayer graphene (TBG) gives rise to a variety of correlated phases such as correlated insulators, superconductivity, orbital ferromagnetism, Chern insulators and nematicity. Most of these phases occur at or near an integer number of carriers per moiré unit cell. Experimental demonstration of ordered states at fractional moiré band-fillings at zero applied magnetic field $B$, is a challenging pursuit. In this letter, we report the observation of states near half-inte…
The broken symmetry that develops below 17.5K in the heavy fermion compound URu2Si2 has long eluded identification. Here we argue that the recent observation of Ising quasiparticles in URu2Si2 results from a spinor hybridization order parameter that breaks double time-reversal symmetry by mixing states of integer and half-integer spin. Such "hastatic order" (hasta:[Latin]spear) hybridizes Kramers conduction electrons with Ising, non-Kramers 5f2 states of the uranium atoms to produce Ising quasiparticles. The development of a spinorial hybridization at 17.5K accounts for both the large entropy …
The interplay of structural and electronic phases in iron-based superconductors is a central theme in the search for the superconducting pairing mechanism. While electronic nematicity, defined as the breaking of four-fold symmetry triggered by electronic degrees of freedom, is competing with superconductivity, the effect of purely structural orthorhombic order is unexplored. Here, using x-ray diffraction (XRD), we reveal a new structural orthorhombic phase with an exceptionally high onset temperature ($T_\mathrm{ort} \sim 250$ K), which coexists with superconductivity ($T_\mathrm{c} = 25$ K), …
Electronic nematicity has been commonly observed in juxtaposition with unconventional superconductivity. Understanding the nature of the nematic state, as well as its consequence on the electronic band structure and superconductivity, has become a pivotal focus in condensed matter physics. Here we use spectroscopic imaging-scanning tunneling microscopy to visualize how the interacting quasiparticles organize themselves in the nematic state of kagome superconductor CsV$_{3-x}$Ti$_x$Sb$_5$, in which twofold symmetric (C$_2$) quasiparticle scattering interference of the vanadium kagome bands emer…
We report the results of the parquet renormalization group (RG) analysis of the phase diagram of the most general 5-pocket model for Fe-based superconductors. We use as an input the orbital structure of excitations near the five pockets made out of $d_{xz}$, $d_{yz}$, and $d_{xy}$ orbitals and argue that there are 40 different interactions between low-energy fermions in the orbital basis. All interactions flow under RG, as one progressively integrates out fermions with higher energies. We find that the low-energy behavior is amazingly simple, despite the large number of interactions. Namely, a…
An electronic nematic phase can be classified by a spontaneously broken discrete rotational symmetry of a host lattice. In a square lattice, there are two distinct nematic phases. The parallel nematic phase breaks $x$ and $y$ symmetry, while the diagonal nematic phase breaks the diagonal $(x+y)$ and anti-diagonal $(x-y)$ symmetry. We investigate the interplay between the parallel and diagonal nematic orders using mean field theory. We found that the nematic phases compete with each other, while they coexist in a finite window of parameter space. The quantum critical point between the diagonal …
Motivated by recent observation of nematicity in Moiré systems, we study three different orbital orders that potentially can happen in Moiré systems: (1) the nematic order; (2) the valley polarization; and (3) the "compass order". Each order parameter spontaneously breaks part of the spatial symmetries of the system. We explore physics caused by the quantum fluctuations close to the order-disorder transition of these order parameters. Especially, we recognize that the symmetry of the Moiré systems leads to a crucial difference of the effective theory describing the nematic order from the stand…
