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We investigate some possible symmetries of the superconducting state that emerges in three-dimensional altermagnets in the presence of spin-orbit coupling. We demonstrate within a weak-coupling approach that these altermagnets, which naturally possess an order modulated by a vector form factor $\boldsymbol{g}_{\mathbf{k}}$, favor spin-triplet superconductivity described by gap functions given by $\boldsymbol{d}(\mathbf{k}) = \boldsymbol{u}(\mathbf{k}) \times \boldsymbol{g}_{\mathbf{k}}$, where $\boldsymbol{u}(\mathbf{k}) = - \boldsymbol{u}(-\mathbf{k})$. Consequently, this singles out $f$-wave spin-triplet superconductivity as the most favorable pairing state to appear in the vicinity of $d$-wave altermagnetism. Furthermore, we obtain that the combination of spin-singlet superconducting states with altermagnetism gives rise to Bogoliubov-Fermi surfaces, which are protected by a $\mathbb{Z}_2$ topological invariant. Using a Ginzburg-Landau analysis, we show that, for a class of spin-orbit coupled altermagnetic models, a superconducting phase is expected to appear at low temperatures as an intertwined $d + if$ state, thus breaking time-reversal symmetry spontaneously., Comment: 7 pages, 3 figures; Supplemental Material: 9 pages, 3 figures

Nematic phases, where electrons in a solid spontaneously break rotational symmetry while preserving the translational symmetry, exist in several families of unconventional superconductors [1, 2]. Although superconductivity mediated by nematic fluctuations is well established theoretically [3-7], it has yet to be unambiguously identified experimentally [8, 9]. A major challenge is that nematicity is often intertwined with other degrees of freedom, such as magnetism and charge order. The FeSe$_{1-x}$S$_x$ family of iron based superconductors provides a unique opportunity to explore this concept, as it features an isolated nematic phase that can be suppressed by sulfur substitution at a quantum critical point (QCP) near $x_c = 0.17$, where nematic fluctuations are the largest [10-12]. Here, we performed scanning tunneling spectroscopy measurements to visualize Boguliubov quasiparticle interference patterns, from which we determined the momentum structure of the superconducting gap near the Brillouin zone $\Gamma$ point of FeSe$_{0.81}$S$_{0.19}$. The results reveal an anisotropic, near nodal gap with minima that are $45^\circ$ rotated with respect to the Fe-Fe direction, characteristic of a nematic pairing interaction, contrary to the usual isotropic gaps due to spin mediated pairing in other tetragonal Fe-based superconductors. The results are also in contrast with pristine FeSe, where the pairing is mediated by spin fluctuations and the gap minima are aligned with the Fe-Fe direction. Therefore, the measured gap structure demonstrates not only a fundamental change of the pairing mechanism across the phase diagram of FeSe$_{1-x}$S$_x$, but it also indicates the existence of superconductivity mediated by nematic fluctuations in FeSe$_{0.81}$S$_{0.19}$.

In an altermagnet, the symmetry that relates configurations with flipped magnetic moments is a rotation. This makes it qualitatively different from a ferromagnet, where no such symmetry exists, or a collinear antiferromagnet, where this symmetry is a lattice translation. In this paper, we investigate the impact of the crystalline environment, enabled by the spin-orbit coupling, on the magnetic and electronic properties of an altermagnet. We find that, because each component of the magnetization acquires its own angular dependence, the Zeeman splitting of the bands has symmetry-protected nodal lines residing on mirror planes of the crystal. Upon crossing the Fermi surface, these nodal lines give rise to pinch points that behave as single or double type-II Weyl nodes. We show that an external magnetic field perpendicular to these mirror planes can only move the nodal lines, such that a critical field value is necessary to collapse the nodes and make the Weyl pinch points annihilate. This unveils the topological nature of the transition from a nodal to a nodeless Zeeman splitting of the bands. We also classify the altermagnetic states of common crystallographic point groups in the presence of spin-orbit coupling, revealing that a broad family of magnetic orthorhombic perovskites can realize altermagnetism., Comment: revised manuscript accepted for publication as an Editors' Suggestion in PRB

We study an exactly solvable model with bond-directional quadrupolar and octupolar interactions between spin-orbital entangled $j_{\mathrm{eff}} = \frac{3}{2}$ moments on the honeycomb lattice. We show that this model features a multipolar spin liquid phase with gapless fermionic excitations. In the presence of perturbations that break time-reversal and rotation symmetries, we find Abelian and non-Abelian topological phases in which the Chern number evaluates to $0$, $\pm 1$, and $\pm 2$. We also investigate quantum phase transitions out of the multipolar spin liquid using a parton mean-field approach and orbital wave theory. In the regime of strong integrability-breaking interactions, the multipolar spin liquid gives way to ferroquadrupolar-vortex and antiferro-octupolar ordered phases that harbor a hidden spin-$\frac{1}{2}$ Kitaev spin liquid. Our work unveils mechanisms for unusual multipolar orders and quantum spin liquids in Mott insulators with strong spin-orbit coupling., Comment: 14 pages, 9 figures

We study the pairing instability of a two-dimensional metallic system induced by Ising-nematic quantum fluctuations in the presence of an unavoidable relevant coupling of the nematic order parameter to the elastic modes (acoustic phonons) of the lattice. We find that this nematoelastic coupling $\lambda_\mathrm{latt}$ leads to a decrease of both the superconducting (SC) critical temperature $T_c$ and the gap function $\Delta$, regardless of the gap symmetry. Interestingly, we show that $\lambda_\mathrm{latt}$ provides a knob that allows us to investigate the emergence of the SC phase at low temperatures, as an instability from either a non-Fermi liquid or a Fermi liquid normal state. The phase transitions between the SC and these normal states are characterized by different critical exponents, which may also vary for each gap symmetry. Finally, we argue that these results might explain the dependence of $T_c$ in the vicinity of the nematic quantum critical point exhibited by the compound FeSe$_{1 - x}$S$_x$., Comment: 6 pages, 4 figures

We investigate the properties of the Kitaev-Kondo lattice model defined on a bilayer honeycomb lattice by means of the SO(3) Majorana representation for spin-$1/2$ moments. We first consider the pairing of neighboring sites for the parent Kitaev spin liquid (KSL) Hamiltonian to render the Majorana and the spin-$1/2$ Hilbert spaces perfectly equivalent to each other. As a consequence, we demonstrate that this decoupling of the Kitaev interaction in terms of the SO(3) Majorana fermions reproduces exactly the spectrum of the KSL model alone. Then, by considering the effect of a local Kondo coupling $J_K$ in the model and decoupling it in terms of an order parameter that physically must have a finite staggering phase, we obtain that the system undergoes a quantum phase transition from a fractionalized Fermi liquid to a nematic triplet superconducting (SC) phase as $J_K$ is increased. Depending on the model parameters, this SC phase can exhibit either Dirac points, Bogoliubov-Fermi lines, or Bogoliubov-Fermi surfaces as nodal bulk manifolds. The surface states in this latter case are also characterized by topologically protected antichiral edge modes. The SC phase breaks time-reversal symmetry and exhibits a coexistence of a dominant odd-frequency pairing with a small even-frequency component for electronic excitations localized on sites of the same sublattice of the system. Finally, we show that this SC phase is in fact a pair-density-wave state, with Cooper pairs possessing a finite center-of-mass momentum in zero magnetic field., Comment: $14 + \epsilon$ pages, 9 figures

Recent experiments in unconventional superconductors, and in particular iron-based materials, have reported evidence of an antiferromagnetic quantum critical point (AFM-QCP) emerging inside the superconducting dome of the phase diagram. Fluctuations associated with such an AFM-QCP are expected to promote unusual temperature dependencies of thermodynamic quantities. Here, we compute the $T$ dependence of the specific heat $C(T)$ deep inside a fully gapped $s^{+-}$ superconducting state as the AFM-QCP is approached. We find that, at the AFM-QCP, the specific heat $C(T)$ vanishes quadratically with temperature, as opposed to the typical exponential suppression seen in fully-gapped BCS superconductors. This robust result is due to a non-analytic contribution to the free-energy arising from the general form of the bosonic (AFM) propagator in the SC state. Away from the AFM-QCP, as temperature is lowered, $C(T)$ shows a crossover from a $T^2$ behavior to an exponential behavior, with the crossover temperature scale set by the value of the superconducting gap and the distance to the QCP. We argue that these features in the specific heat can be used to unambiguously determine the existence of AFM-QCPs inside the superconducting domes of iron-based and other fully gapped unconventional superconductors., Comment: 10 pages, 5 figures, new Section V included

We propose an exactly solvable model for $j_{\text{eff}}=\frac32$ local moments on the honeycomb lattice. Our construction is guided by a symmetry analysis and by the requirement of an exact solution in terms of a Majorana fermion representation for multipole operators. The main interaction in the model can be interpreted as a bond-dependent quadrupole-quadrupole interaction. When time reversal symmetry is explicitly broken, we obtain a gapped spin liquid with a single chiral Majorana edge mode. We also investigate another solvable model in which the time-reversal-invariant spin liquid is coupled to conduction electrons in a superconductor. In the presence of a Kondo-like coupling that involves the octupole moment of the localized spins, the itinerant electrons hybridize with the emergent Majorana fermions in the spin liquid. This leads to spontaneous time reversal symmetry breaking and generates odd-frequency pairing. Our results suggest that $j_{\text{eff}}=\frac32$ systems with strong quadrupole-quadrupole interactions may provide a route towards non-Abelian quantum spin liquids and unconventional superconductivity., Comment: 16 pages, 8 figures. We corrected the statement about the number of edge states in the chiral phase of the quadrupolar spin liquid. We also added the appendix B

We calculate the resistivity associated with an Ising-nematic quantum critical point in the presence of disorder and acoustic phonons in the lattice model. We use the memory-matrix transport theory, which has a crucial advantage compared to other methods of not relying on the existence of well-defined quasiparticles in the low-energy effective theory. As a result, we obtain that by including an inevitable interaction between the nematic fluctuations and the elastic degrees of freedom of the lattice (parametrized by the nemato-elastic coupling $\kappa_{\text{latt}}$), the resistivity $\rho(T)$ of the system as a function of temperature obeys a universal scaling form described by $\rho(T)\sim T\ln (1/T)$ at high temperatures, reminiscent of the paradigmatic strange metal regime observed in many strongly correlated compounds. For a window of temperatures comparable with $\kappa^{3/2}_{\text{latt}}\varepsilon_F$ (where $\varepsilon_F$ is the Fermi energy of the microscopic model), the system displays another regime in which the resistivity is consistent with a description in terms of $\rho(T)\sim T^{\alpha}$, where the effective exponent roughly satisfies the inequality $1\lesssim\alpha\lesssim 2$. However, in the low-temperature limit (i.e., $T\ll\kappa^{3/2}_{\text{latt}}\varepsilon_F$), the properties of the quantum critical state change in an important way depending on the types of disorder present in the system: It can either recover a conventional Fermi liquid described by $\rho(T)\sim T^2$ or it could exhibit yet another non-Fermi liquid regime characterized by the scaling form $\rho(T)-\rho_0\sim T^2\ln T$. Our results emphasize the key role played by both phonon and disorder effects in the scenario of nematic quantum criticality and might be fundamental for addressing recent transport experiments in some iron-based superconductors., Comment: 18 pages, 9 figures

We investigate the strong influence of the $\Theta_{II}$-loop-current order on both unidirectional and bidirectional d-wave charge-density-wave/pair-density-wave (CDW/PDW) composite orders along axial momenta $(\pm Q_0,0)$ and $(0,\pm Q_0)$ that emerge in an effective hot spot model departing from the three-band Emery model relevant to the phenomenology of the cuprate superconductors. This study is motivated by the compelling evidence that the $\Theta_{II}$-loop-current order described by this model may explain groundbreaking experiments such as spin-polarized neutron scattering performed in these materials. Here, we demonstrate, within a saddle-point approximation, that the $\Theta_{II}$-loop-current order clearly coexists with bidirectional (i.e. checkerboard) d-wave CDW and PDW orders along axial momenta, but is visibly detrimental to the unidirectional (i.e. stripe) case. This result has potentially far-reaching implications for the physics of the cuprates and agrees well with very recent x-ray experiments on YBCO that indicate that at higher dopings the CDW order has indeed a tendency to be bidirectional., Comment: Published in Physical Review B

We study the fate of the so-called $\Theta_{II}$-loop-current order that breaks both time-reversal and parity symmetries in a two-dimensional hot spot model with antiferromagnetically mediated interactions, using Fermi surfaces relevant to the phenomenology of the cuprate superconductors. We start from a three-band Emery model describing the hopping of holes in the CuO$_{2}$ plane that includes two hopping parameters $t_{pp}$ and $t_{pd}$, local on-site Coulomb interactions $U_{d}$ and $U_{p}$ and nearest-neighbor $V_{pd}$ couplings between the fermions in the copper [Cu$(3d_{x^{2}-y^{2}})$] and oxygen [O$(2p_{x})$ and O$(2p_{y})$] orbitals. By focusing on the lowest-energy band, we proceed to decouple the local interaction $U_{d}$ of the Cu orbital in the spin channel using a Hubbard-Stratonovich transformation to arrive at the interacting part of the so-called spin-fermion model. We also decouple the nearest-neighbor interaction $V_{pd}$ to introduce the order parameter of the $\Theta_{II}$-loop-current order. In this way, we are able to construct a consistent mean-field theory that describes the strong competition between the composite order parameter made of a quadrupole-density-wave and $d$-wave pairing fluctuations proposed in Efetov \emph{et al.} [Nat. Phys. \textbf{9}, 442 (2013)] with the $\Theta_{II}$-loop-current order parameter that is argued to be relevant for explaining important aspects of the physics of the pseudogap phase displayed in the underdoped cuprates., Comment: 16 pages, 5 figures. v2: minor revisions, references added. The magnetic moment per unit-cell associated with the $\Theta_{II}$-loop-current-phase is calculated and compared with experimental results. Accepted for publication in Physical Review B

In light of the new experimental and theoretical important developments in high-$T_c$ superconductivity, we revisit the fermionic hot-spot model relevant to the phenomenology of the cuprates. We extend previous results by means of a complete two-loop order renormalization group (RG) framework. Here, we explicitly study the effect of the charge-density-wave (CDW) order parameter with a $d$-wave form factor with the experimentally observed modulation $(\pm Q_0,0)$ and $(0,\pm Q_0)$ at the infrared-stable nontrivial fixed point obtained previously for this model. Additionally, we proceed to investigate also the so-called pair-density-wave (PDW) order that was recently proposed in the literature as a possible candidate for the "hidden" order to describe the pseudogap phase observed in underdoped cuprates. We confirm that although the above two ordering tendencies are also found to be nearly degenerate both at one-loop and two-loop RG orders and linked by an emergent $SU(2)$ pseudospin symmetry, they turn out to be subleading for weaker couplings in the present model to antiferromagnetism, $d$-wave bond-density wave (BDW) order with modulation along Brillouin zone diagonals $(\pm Q_0,\pm Q_0)$, and $d$-wave singlet superconductivity (SSC). However, as we increase the strength of the initial coupling towards moderate values, we do capture a tendency for the entangled PDW/CDW order to become leading compared to BDW/SSC in the model, which suggests that the former composite order might be indeed a viable concept to describe some cuprate superconductors at high temperatures in the underdoped regime, as has been recently alluded to by many authors in the literature., Comment: 12 pages, 6 figures. Published version

The two-loop renormalization group (RG) calculation is considerably extended here for the two-dimensional (2D) fermionic effective field theory model, which includes only the so-called "hot spots" that are connected by the spin-density-wave (SDW) ordering wavevector on a Fermi surface generated by the 2D $t-t'$ Hubbard model at low hole doping. We compute the Callan-Symanzik RG equation up to two loops describing the flow of the single-particle Green's function, the corresponding spectral function, the Fermi velocity, and some of the most important order-parameter susceptibilities in the model at lower energies. As a result, we establish that -- in addition to clearly dominant SDW correlations -- an approximate (pseudospin) symmetry relating a short-range \emph{incommensurate} $d$-wave charge order to the $d$-wave superconducting order indeed emerges at lower energy scales, which is in agreement with recent works available in the literature addressing the 2D spin-fermion model. We derive implications of this possible electronic phase in the ongoing attempt to describe the phenomenology of the pseudogap regime in underdoped cuprates., Comment: 16 pages, 7 figures. Published version

Motivated by a recent experimental observation of a nodal liquid on both single crystals and thin films of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ by Chatterjee \emph{et al.} [Nature Physics \textbf{6}, 99 (2010)], we perform a field-theoretical renormalization group (RG) analysis of a two-dimensional model consisting of eight points located near the "hot spots" on the Fermi surface which are directly connected by spin density wave ordering wave vector. We derive RG equations up to two-loop order describing the flow of renormalized couplings, quasiparticle weight, several order-parameter response functions, and uniform spin and charge susceptibilities of the model. We find that while the order-parameter susceptibilities investigated here become non-divergent at two loops, the quasiparticle weight vanishes in the low-energy limit, indicating a breakdown of Fermi liquid behavior at this RG level. Moreover, both uniform spin and charge susceptibilities become suppressed in the scaling limit which indicate gap openings in both spin and charge excitation spectra of the model., Comment: 11 pages, 11 figures. Minor revision, typos corrected, references added

We perform a renormalization group (RG) study up to two-loop order of an effective low-energy two-band model to describe some of the recently discovered iron-based superconductors. Our starting point is the itinerant electronic model proposed by Chubukov \emph{et al.} [Phys. Rev. B \textbf{78}, 134512 (2008)], which displays two small, almost nested Fermi pockets with one hole pocket centered at $(0,0)$ and one electron pocket centered at $\mathbf{Q} = (\pi,\pi)$ in the folded Brillouin zone. We then proceed to implement a complete two-loop RG calculation for this model of four-point vertex corrections, quasiparticle weight and several order-parameter susceptibilities in order to evaluate the robustness of one-loop RG results available in the literature with respect to including self-energy effects and higher-order quantum fluctuations., Comment: 6 pages, 6 figures; accepted for publication in Europhysics Letters