The interplay of competing orders is relevant to high-temperature superconductivity known to emerge upon suppression of a parent antiferromagnetic order typically via charge doping. How such interplay evolves at low temperature---in particular at what doping level the zero-temperature quantum critical point (QCP) is located---is still elusive because it is masked by the superconducting state. The QCP had long been believed to follow a smooth extrapolation of the characteristic temperature $T^*$ for the strange normal state well above the superconducting transition temperature. However, recently the $T^*$ within the superconducting dome was reported to unexpectedly exhibit back-bending likely in the cuprate Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+δ}$. Here we show that the original and revised phase diagrams can be understood in terms of weak and moderate competitions, respectively, between superconductivity and a pseudogap state such as $d$-density-wave or spin-density-wave, based on both Ginzburg-Landau theory and the realistic $t$-$t^{\prime}$-$t^{\prime\prime}$-$J$-$V$ model for the cuprates. We further found that the calculated temperature and doping-level dependence of the quasiparticle spectral gap and Raman response qualitatively agrees with the experiments. In particular, the $T^*$ back-bending can provide a simple explanation of the observed anomalous two-step thermal evolution dominated by the superconducting gap and the pseudogap, respectively. Our results imply that the revised phase diagram is likely to take place in high-temperature superconductors.
We present a theoretical framework for a quantitative understanding of the full doping dependence of the optical spectra of the cuprates. In accord with experimental observations, the computed spectra show how the high-energy Mott features continue to persist in the overdoped regime even after the mid-infrared (MIR) peak originating from the pseudogap has collapsed in a quantum critical point. In this way, we reconcile the opposing tendencies of the MIR and Mott peaks to shift in opposite directions in the optical spectra with increasing doping. The competition between the pseudogap and the Mo…
We discuss the problem of competition between a superconducting (SC) ordered state with a charge density wave (CDW) state in stripe phases of high $T_c$ superconductors. We consider an effective model for each stripe motivated by studies of spin-gapped electronic ladder systems. We analyze the problem of dimensional crossover arising from inter-stripe SC and CDW couplings using non-Abelian bosonization and renormalization group (RG) arguments to derive an effective $O(4)$-symmetric nonlinear $σ$-model in $D=2+1$ for the case of when both inter-stripe couplings are of equal magnitude as well as…
Third-harmonic generation in superconductors: Role of quantum geometry in the competition between Higgs mode and quasiparticles
The University of Tokyo（东京大学）；The Hong Kong University of Science and Technology（香港科技大学）；RIKEN Center for Emergent Matter Science (CEMS)（理化学研究所凝聚体物质科学中心）
We show that, under rather general assumptions, the phase diagram of a quasi-one-dimensional repulsive Fermi system consists of two ordered phases: the density wave, spin or charge, and the superconductivity. It is demonstrated that the symmetry of the superconducting order parameter is a non-universal property sensitive to microscopic details of the model. Three potentially stable superconducting states are identified: they are triplet $f$-wave, singlet $d_{x^2-y^2}$-wave, and $d_{xy}$-wave. Presence of multiple competing superconducting states implies that for a real material this symmetry i…
In topological insulators doped with magnetic ions, spin-orbit coupling and ferromagnetism give rise to the quantum anomalous Hall effect. Here we show that in s-wave superconductors with strong spin-orbit coupling, magnetic impurity ions can generate topological vortices in the absence of external magnetic fields. Such vortices, dubbed quantum anomalous vortices, support robust Majorana zero-energy modes when superconductivity is induced in the topological surface states. We demonstrate that the zero-energy bound states observed in Fe(Te,Se) superconductors are possible realizations of the Ma…
A microscopic theory for the spin triplet Cooper pairing in non-centrosymmetric superconductors like CePt_3Si and CeTSi_3 (T=Rh, Ir) is presented. The lack of inversion symmetry leads to new anomalous spin fluctuations which stabilize the triplet part in addition to the singlet part originating from the centrosymmetric spin fluctuations. It is shown that both parts have similar nontrivial momentum dependence of A_1 type. Therefore the mixed singlet-triplet gap function has accidental line nodes on both Fermi surface sheets which are stable as function of temperature. This gap function explains…
We investigate the anomalous flux state of interacting higher-order topological superconductors (HOTSC) protected by rotation symmetries. By introducing a $π$ superconducting flux in 2D HOTSC, we demonstrate the existence of a robust zero mode trapped at the flux center. Remarkably, the rotation symmetry and fermion parity display projective representation inside the $π$ flux with $N=2$ supersymmetry algebra. A similar gapless flux pattern also exists in 3D HOTSCs, the flux lines of which carry anomalous helical modes that cannot be realized on purely one-dimensional lattice models. Notably, t…
The normal state behavior in the heavy-fermion superconductor Ce2PdIn8 has been probed by means of Hall coefficient (RH) and transverse magnetoresistivity (MR) measurements. The results indicate the predominance of contributions from antiferromagnetic spin fluctuations at low temperatures. Anomalous non-Fermi-liquid-like features, observed below 8 K in both RH(T) and MR(T), are related to underlying quantum critical point, evidenced before in the specific heat and the electrical resistivity data. The magnetotransport in Ce2PdIn8 is shown to exhibit specific types of scaling that may appear uni…
Unconventional superconductors are most frequently identified by the observation of power-law behaviour on low-temperature thermodynamic or transport properties, such as specific heat. Here we show that, in addition to the usual point and line nodes, a much wider class of different nodal types can occur. These new types of nodes typically occur when there are transitions between different types of gap node topology, for example when point or line nodes first appear as a function of some physical parameter. We identify anomalous, non-integer thermodynamic power laws associated with these new no…
Recent findings of anomalous super-linear scaling of low temperature ($T$) penetration depth (PD) in several nodal superconductors near putative quantum critical points suggest that the low temperature PD can be a useful probe of quantum critical fluctuations in a superconductor. On the other hand, cuprates which are poster child nodal superconductors have not shown any such anomalous scaling of PD, despite growing evidence of quantum critical points. Then it is natural to ask when and how can quantum critical fluctuations cause anomalous scaling of PD? Carrying out the renormalization group c…
The role of electron-phonon interactions in iron-based superconductor is currently under debate with conflicting experimental reports on the isotope effect. To address this important issue, we employ the renormalization-group method to investigate the competition between electron-electron and electron-phonon interactions in these materials. The renormalization-group analysis shows that the ground state is a phonon-dressed unconventional superconductor: the dominant electronic interactions account for pairing mechanism while electron-phonon interactions are subdominant. Because of the phonon dr…
The coexistence state of antiferromagnetism (AF) and superconductivity (SC) has been observed in five-layered cuprates. However, this coexistence state disappears, and the AF phase and SC phase lose contact in the doping phase diagram toward double- and single-layered cuprates. We investigate the mechanism of the disappearance of the coexistence of AF and SC in disordered cuprate superconductors in order to understand these doping phase diagrams. In single- and double-layered cuprates, electrons on the CuO$_2$ plane experience the disorder effect through inhomogeneity in the charge reservoir l…
We identify an intrinsic mechanism of the anomalous Hall effect for non-symmorphic chiral superconductors. This mechanism relies on both a nontrivial multi-band chiral superconducting order parameter, which is a mixture of pairings of even and odd angular momentum channels, and a complex normal state inter-sublattice hopping, both of which are consequences of the nonsymmorphic group symmetry of the underlying lattice. We apply this mechanism to the putative chiral superconducting phase of the heavy-fermion superconductor $\mathrm{UPt_3}$ and calculate the anomalous ac Hall conductivity in a si…
Recent high-precision measurements employing different experimental techniques have unveiled an anomalous peak in the doping dependence of the London penetration depth which is accompanied by anomalies in the heat capacity in iron-pnictide superconductors at the optimal composition associated with the hidden antiferromagnetic quantum critical point. We argue that finite temperature effects can be a cause of observed features. Specifically we show that quantum critical magnetic fluctuations under superconducting dome can give rise to a nodal-like temperature dependence of both specific heat and…
We study proximity effects close to a boundary between s and s+- superconductors. Frustration, caused by interaction of the s-wave gap parameter with the opposite-sign gaps of s+- superconductor, leads to several anomalous features. In the case of strong frustration a nontrivial time-reversal-symmetry breaking (TRSB) state, with nonzero phase angles between all gap parameters, is possible. In a more typical state, the s-wave order parameter is aligned with one of the s+- gaps. The other (anti-aligned) gap induces negative feature in the s-wave density of states, which can serve as a fingerprin…
Upper critical field, H_c2, in quasi-1D superconductors is investigated by the weak coupling renormalization group technique. It is shown that H_c2 greatly exceeds not only the Pauli limit, but also the conventional paramagnetic limit of the Flude-Ferrell-Larkin-Ovchinnikov (FFLO) state. This increase is mainly due to quasi-1D fluctuations effect as triggered by interference between unconventional superconductivity and density-wave instabilities. Our results give a novel viewpoint on the large H_c2 observed in TMTSF-salts in terms of a d-wave FFLO state that is predicted to be verified by the …
To understand the origin of the nodal gap structure realized in BaFe$_2$(As,P)$_2$, we study the three-dimensional gap structure based on the three-dimensional ten-orbital Hubbard model with quadrupole interaction. In this model, strong spin and orbital fluctuations develop by using the random-phase-approximation. By solving the Eliashberg gap equation, we obtain the fully-gapped s-wave state with (without) sign reversal between hole-like and electron-like Fermi surfaces due to strong spin (orbital) fluctuations, so called the $s_\pm$-wave ($s_{++}$-wave) state. When both spin and orbital fluc…
When a second-order magnetic phase transition is tuned to zero temperature by a non-thermal parameter, quantum fluctuations are critically enhanced, often leading to the emergence of unconventional superconductivity. In these `quantum critical' superconductors it has been widely reported that the normal-state properties above the superconducting transition temperature $T_c$ often exhibit anomalous non-Fermi liquid behaviors and enhanced electron correlations. However, the effect of these strong critical fluctuations on the superconducting condensate below $T_c$ is less well established. Here w…
Fluctuations around an antiferromagnetic quantum critical point (QCP) are believed to lead to unconventional superconductivity and in some cases to high-temperature superconductivity. However, the exact mechanism by which this occurs remains poorly understood. The iron-pnictide superconductor BaFe$_2$(As$_{1-x}$P$_x$)$_2$ is perhaps the clearest example to date of a high temperature quantum critical superconductor, and so it is a particularly suitable system in which to study how the quantum critical fluctuations affect the superconducting state. Here we show that the proximity of the QCP yiel…
We investigate the competition between d-wave superconductivity and nematic order in high-$T_c$ superconductor, and examine the role played by gapless fermionic degrees of freedom. Apart from the competitive interaction with the superconducting order parameter, the nematic order parameter couples strongly to gapless nodal quasiparticles. The interplay of these two kinds of interactions is analyzed by means of renormalization group method. In case the fermionic degree of freedom are entirely neglected, the competitive interaction between two bosonic order parameters is strongly relevant, and ca…
Anomalous metallic behavior, marked by a saturating finite resistivity much lower than the Drude estimate, has been observed in a wide range of two-dimensional superconductors. Utilizing the electrostatically gated LaAlO3/SrTiO3 interface as a versatile platform for superconductor-metal quantum phase transitions, we probe variations in the gate, magnetic field, and temperature to construct a phase diagram crossing from superconductor, anomalous metal, vortex liquid, to Drude metal states, combining longitudinal and Hall resistivity measurements. We find that the anomalous metal phases induced …
When periodicity of crystal is disturbed by atomic disorder, its electronic state becomes inhomogeneous and band dispersion is obscured. In case of Fe-based superconductors, disorder of chalcogen/pnictogen height causes disorder of Fe 3d level splitting. Here, we report an angle-resolved photoemission spectroscopy study on FeSe_1-xTe_x with the chalcogen height disorder, showing that the disorder affects the Fe 3d band dispersions in an orbital-selective way instead of simple obscuring effect. The reverse of the Fe 3d level splitting due to the chalcogen height difference causes the splitting …
We present variational Monte Carlo (VMC) results for the Gutzwiller-projected coexisting state including both antiferromagnetic (AFM) order and superconducting (SC) order in the two-dimensional t-t'-t"-J model. By further considering off-site spin correlation between electrons, in contrast to earlier VMC results [Phys. Rev. Lett. 102, 027002 (2009)], we find the apparent competition between AFM order and SC order near the underdoped regime instead of coexistence. The local ferromagnetic correlation introduced by spin-spin Jastrow correlators seem to be responsible for the disappearance of the …
We show that a Weyl superconductor can absorb light via a novel surface-to-bulk mechanism, which we dub the topological anomalous skin effect. This occurs even in the absence of disorder for a single-band superconductor, and is facilitated by the topological splitting of the Hilbert space into bulk and chiral surface Majorana states. In the clean limit, the effect manifests as a characteristic absorption peak due to surface-bulk transitions. We also consider the effects of bulk disorder, using the Keldysh response theory. For weak disorder, the bulk response is reminiscent of the Mattis-Bardee…
We show that the gapless boundary signatures - namely, chiral/helical hinge modes or localized zero modes - of three-dimensional higher-order topological insulators and superconductors with inversion symmetry can be gapped without symmetry breaking upon the introduction of non-Abelian surface topological order. In each case, the fractionalization pattern that appears on surface is `anomalous' in the sense that it can be made consistent with symmetry only on the surface of a three dimensional higher-order insulator/superconductor. Our results show that the interacting manifestation of higher-or…
Clapping modes, which are relative amplitude and phase modes between two chiral components of Cooper pairs, are bosonic collective modes inherent to chiral superconductors. These modes behave as long-lived bosons with masses smaller than the threshold energy, $2|Δ|$, for decay into unbound fermion pairs. Here, we clarify that the real/imaginary clapping modes in chiral superconductors directly couple to acoustic wave propagation when the weak particle-hole asymmetry of the normal state quasiparticle dispersion is taken into account. The clapping modes driven by an acoustic wave generate an alt…
We demonstrate the formation of robust zero energy modes close to magnetic impurities in the iron-based superconductor FeSe$_{1-x}$Te$_x$. We find that the Zeeman field generated by the impurity favors a spin-triplet inter-orbital pairing as opposed to the spin-singlet intra-orbital pairing prevalent in the bulk. The preferred spin-triplet pairing preserves time-reversal symmetry and is topological, as robust, topologically-protected zero modes emerge at the boundary between regions with different pairing states. Moreover, the zero modes form Kramers doublets that are insensitive to the direct…
We propose to realize a two-dimensional chiral topological superconducting (TSC) state from the quantum anomalous Hall plateau transition in a magnetic topological insulator thin film through the proximity effect to a conventional $s$-wave superconductor. This state has a full pairing gap in the bulk and a single chiral Majorana mode at the edge. The optimal condition for realizing such chiral TSC is to have inequivalent superconducting pairing amplitudes on top and bottom surfaces of the doped magnetic topological insulator. We further propose several transport experiments to detect the chira…
A weak superconducting proximity effect in the vicinity of the topological transition of a quantum anomalous Hall system has been proposed as a venue to realize a topological superconductor (TSC) with chiral Majorana edge modes (CMEMs). A recent experiment [Science 357, 294 (2017)] claimed to have observed such CMEMs in the form of a half-integer quantized conductance plateau in the two-terminal transport measurement of a quantum anomalous Hall-superconductor junction. Although the presence of a superconducting proximity effect generically splits the quantum Hall transition into two phase tran…
We examine the transport properties of a double quantum dot system coupled to a topological superconducting nanowire hosting Majorana quasiparticles at its ends, with the central quantum dot attached to the left and right leads. We focus on the behavior of the local density of states and the linear conductance, calculated with the aid of the numerical renormalization group method, to describe the influence of the Majorana coupling on the low-temperature transport properties induced by the Kondo correlations. In particular, we show that the presence of Majorana quasiparticles in the system affe…
Anomalous transport behavior -- both longitudinal and Hall -- is the defining characteristic of the strange-metal state of High-Tc cuprates. The temperature, frequency, and magnetic field dependence of the resistivity is understood within strange metal phenomenology as resulting from energy-scale competition to set the inelastic relaxation rate. The anomalously strong temperature dependence of the Hall coefficient, however, is at odds with this phenomenology. Here we report measurements of the Hall resistivity in the strange metal state of cuprates over a broad range of magnetic fields and tem…
Magnetic competition in topological kagome magnets is studied by incorporating the spin-orbit coupling, the anisotropic Hund coupling and spin exchange into the kagome lattice. Using the Bogoliubov variational principle we find the stable phases at zero and finite temperatures. At zero temperature and in the strong Ising-Hund coupling regime, a magnetic tunability from the out-of-plane ferromagnetism (FM) to the in-plane antiferromagnetism (AFM) is achieved by a universal property of the critical in-plane Hund coupling. At two-thirds filling the phase transition from the out-of-plane FM to the…
