Luttinger semimetals have quadratic band crossings at the Brillouin zone-center in three spatial dimensions. Coulomb interactions in a model that describes these systems stabilize a non-trivial fixed point associated with a non-Fermi liquid state, also known as the Luttinger-Abrikosov-Beneslavskii phase. We calculate the optical conductivity $σ(ω) $ and the dc conductivity $σ_{dc} (T) $ of this phase, by means of the Kubo formula and the Mori-Zwanzig memory matrix method, respectively. Interestingly, we find that $σ(ω) $, as a function of the frequency $ω$ of an applied ac electric field, is characterized by a small violation of the hyperscaling property in the clean limit, which is in marked contrast to the low-energy effective theories that possess Dirac quasiparticles in the excitation spectrum and obey hyperscaling. Furthermore, the effects of weak short-ranged disorder on the temperature-dependence of $σ_{dc} (T)$ give rise to a much stronger power-law suppression at low temperatures compared to the clean limit. Our findings demonstrate that these disordered systems are actually power-law insulators. Our theoretical results agree qualitatively with the data from recent experiments performed on Luttinger semimetal compounds like the pyrochlore iridates [ (Y$_{1-x}$Pr$_x$)$_2$Ir$_2$O$_7$ ].
We study both noncentrosymmetric and time-reversal breaking Weyl semimetal systems under a strong magnetic field with the Coulomb interaction. The three-dimensional bulk system is reduced to many mutually interacting quasi-one-dimensional wires. Each strongly correlated wire can be approached within the Tomonaga-Luttinger liquid formalism. Including impurity scatterings, we inspect the localization effect and the temperature dependence of the electrical resistivity. The effect of a large number of Weyl points in real materials is also discussed.
We show that hybrid Dirac and Weyl semimetals can be realized in a three-dimensional Luttinger semimetal with quadratic band touching (QBT). We illustrate this using periodic kicking scheme. In particular, we focus on a momentum-dependent drivings (nonuniform driving) and demonstrate the realization of various hybrid Dirac and Weyl semimetals. We identify a unique hybrid dispersion Dirac semimetal with two nodes, where one of the nodes is linear while the other is dispersed quadraticlly. Next, we show that by tilting QBT via periodic driving and in the presence of an external magnetic field, o…
We study the interaction between elliptically polarized light and a three-dimensional Luttinger semimetal with quadratic band touching using Floquet theory. In the absence of light, the touching bands can have the same or the opposite signs of the curvature; in each case, we show that simply tuning the light parameters allows us to create a zoo of Weyl semimetallic phases. In particular, we find that double and single Weyl points can coexist at different energies, and they can be tuned to be type I or type II. We also find an unusual phase transition, in which a pair of Weyl nodes form at fini…
Due to strong spin-orbit coupling, charge carriers in three-dimensional quadratic band touching Luttinger semimetals have non-trivial wavefunctions characterized by a pseudospin of 3/2. We compute the dielectric permittivity of such semimetals at finite doping, within the random phase approximation. Because of interband coupling, the dielectric screening shows a reduced plasma frequency and an increased spectral weight at small wavevectors, compared to a regular quadratic band. This weakens the effective Coulomb potential, modifying the single-particle self-energy for which we present both ana…
Luttinger semimetals are three-dimensional electron systems with a parabolic band touching and an effective total spin $J=3/2$. In this paper, we present an analytical theory of dielectric screening of inversion-symmetric Luttinger semimetals with an arbitrary carrier density and conduction-valence effective mass asymmetry. Assuming a spherical approximation for the single-particle Luttinger Hamiltonian, we determine analytically the dielectric screening function in the random phase approximation for arbitrary values of the wave vector and frequency, the latter in the complex plane. We use thi…
We study a simple model of three-dimensional fermions close to a quadratic band touching point, built from the celebrated Luttinger single-particle Hamiltonian and an attractive contact interaction between the particles. Such a system displays a novel quantum critical point between the semimetallic and an s-wave superconducting phase at which the low-energy "Luttinger fermions" are inextricably coupled to the order parameter fluctuations. The quantum critical point is perturbatively accessible near four spatial dimensions, where it features nontrivial scaling with dynamical exponent 1<z<2 and …
Superconducting instability can occur in three-dimensional quadratic band crossing semimetals only at a finite coupling strength, due to the vanishing of density of states at the quadratic band touching point. Since realistic materials are always disordered to some extent, we study the effect of short-ranged correlated disorder on this superconducting quantum critical point, using a controlled loop-expansion applying dimensional regularization. The renormalization group (RG) scheme allows us to determine the RG flows of the various interaction strengths, and shows that disorder destroys the su…
Luttinger semimetals include materials like gray tin ($α$-Sn) and mercury telluride, which are three-dimensional gapless semiconductors having a quadratic band crossing point (QBCP). Due to a growing interest in QBCPs and new experimental efforts, it is essential to study the finite-temperature properties of such systems. In this paper, we investigate the emergence of plasmons in the presence of Coulomb interactions in isotropic Luttinger semimetals, for zero as well as generic nonzero temperatures. When the Fermi level lies right at the QBCP, which is the point where twofold degenerate conduc…
The surface of a Weyl semimetal famously hosts an exotic topological metal that contains open Fermi arcs rather than closed Fermi surfaces. In this work, we show that the surface is also endowed with a feature normally associated with strongly interacting systems, namely, Luttinger arcs, defined as zeros of the electron Green's function. The Luttinger arcs connect surface projections of Weyl nodes of opposite chirality and form closed loops with the Fermi arcs when the Weyl nodes are undoped. Upon doping, the ends of the Fermi and Luttinger arcs separate and the intervening regions get filled …
We study superconductivity driven by screened Coulomb repulsion in three-dimensional Luttinger semimetals with a quadratic band touching and strong spin-orbit coupling. In these semimetals, the Cooper pairs are formed by spin-3/2 fermions with non-trivial wavefunctions. We numerically solve the linear Eliashberg equation to obtain the critical temperature of a singlet s-wave gap function as a function of doping, with account of spin-orbit and self-energy corrections. In order to understand the underlying mechanism of superconductivity, we compute the sensitivity of the critical temperature to …
We investigate the superconductivity of 3D Luttinger semimetals, such as YPtBi, where Cooper pairs are constituted of spin-3/2 quasiparticles. Various pairing mechanisms have already been considered for these semimetals, such as from polar phonons modes, and in this work we explore pairing from the screened electron-electron Coulomb repulsion. In these materials, the small Fermi energy and the spin-orbit coupling strongly influence how charge fluctuations can mediate pairing. We find the superconducting critical temperature as a function of doping for an s-wave order parameter, and determine i…
Luttinger's theorem is a fundamental result in the theory of interacting Fermi systems: it states that the volume inside the Fermi surface is left invariant by interactions, if the number of particles is held fixed. Although this is traditionally justified using perturbation theory, it can be viewed as arising from a momentum balance argument that examines the response of the ground state to the insertion of a single flux quantum [M. Oshikawa, Phys. Rev. Lett. 84, 3370 (2000)]. This reveals that the Fermi sea volume is a topologically protected quantity. Extending this approach, I show that sp…
Dirac and Weyl semimetals with linearly crossing bands are the focus of much recent interest in condensed matter physics. Although they host fascinating phenomena, their physics can be understood in terms of weakly interacting electrons. In contrast, more than 40 years ago, Abrikosov pointed out that quadratic band touchings are generically strongly interacting. We have performed terahertz spectroscopy on films of the conducting pyrochlore Pr$_2$Ir$_2$O$_7$, which has been shown to host a quadratic band touching. A dielectric constant as large as $\tilde{\varepsilon }/ε_0 \sim 180 $ is observe…
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 fe…
Topological superconductivity in multiband systems has received much attention due to a variety of possible exotic superconducting order parameters as well as non-trivial bulk and surface states. While the impact of coexisting magnetic order on superconductivity has been studied for many years, such as ferromagnetic superconductors, the implication of coexisting multipolar order has not been explored much despite the possibility of multipolar hidden order in a number of $f$-electron materials. In this work, we investigate topological properties of multipolar superconductors that may arise when…
We illuminate the intriguing role played by spatial anisotropy in three-dimensional Luttinger semimetals featuring quadratic band touching and long-range Coulomb interactions. We observe the anisotropy to be subject to an exceptionally slow renormalization group (RG) evolution so that it can be considered approximately constant when computing the impact of quantum fluctuations on the remaining couplings of the system. Using perturbative RG we then study the competition of all local short-range interactions that are generated from the long-range interactions for fixed anisotropy. Two main effec…
Luttinger's theorem connects a basic microscopic property of a given metallic crystalline material, the number of electrons per unit cell, to the volume, enclosed by its Fermi surface, which defines its low-energy observable properties. Such statements are valuable since, in general, deducing a low-energy description from microscopics, which may perhaps be regarded as the main problem of condensed matter theory, is far from easy. In this paper we present a unified framework, which allows one to discuss Luttinger theorems for ordinary metals, as well as closely analogous exact statements for to…
We study the band structure topology and engineering from the interplay between local moments and itinerant electrons in the context of pyrochlore iridates. For the metallic iridate Pr$_2$Ir$_2$O$_7$, the Ir $5d$ conduction electrons interact with the Pr $4f$ local moments via the $f$-$d$ exchange. While the Ir electrons form a Luttinger semimetal, the Pr moments can be tuned into an ordered spin ice with a finite ordering wavevector, dubbed "Melko-Hertog-Gingras" state, by varying Ir and O contents. We point out that the ordered spin ice of the Pr local moments generates an internal magnetic …
We consider the three-dimensional spin-orbit-coupled Luttinger semimetal of "spin" $3/2$ particles in presence of weak attractive interaction in the $l=1$ (p-wave) channel, and determine the low-temperature phase diagram for both particle- and hole-dopings. The phase diagram depends crucially on the sign of the chemical potential, with two different states (with total angular momentum $j=0$ and $j=3$) competing on the hole-doped side, and three (one $j=1$ and two different $j=2$) states on the particle-doped side. The ground state condensates of Cooper pairs with the total angular momentum $j=…
We investigate the optical response properties of three-dimensional Luttinger semimetals with the Fermi energy close to a quadratic band touching point. In particular, in order to address recent experiments on the spectroscopy of Pyrochlore Iridates and half-Heusler superconductors, we derive expressions for the optical conductivity in both the normal and general superconducting states in the linear response regime within the random phase approximation. The response functions can be decomposed into contributions from intraband and interband transitions, the latter comprising a genuine signatur…
We investigate topological Cooper pairing, including gapless Weyl and fully gapped class DIII superconductivity, in a three-dimensional doped Luttinger semimetal. The latter describes effective spin-3/2 carriers near a quadratic band touching and captures the normal-state properties of the 227 pyrochlore iridates and half-Heusler alloys. Electron-electron interactions may favor non-$s$-wave pairing in such systems, including even-parity $d$-wave pairing. We argue that the lowest energy $d$-wave pairings are always of complex (e.g., $d + i d$) type, with nodal Weyl quasiparticles. This implies …
We investigate unconventional superconductivity in three-dimensional electronic systems with the chemical potential close to a quadratic band touching point in the band dispersion. Short-range interactions can lead to d-wave superconductivity, described by a complex tensor order parameter. We elucidate the general structure of the corresponding Ginzburg--Landau free energy and apply these concepts to the case of an isotropic band touching point. For vanishing chemical potential, the ground state of the system is given by the superconductor analogue of the uniaxial nematic state, which features…
We study a system of crossed spin-gapped and gapless Luttinger liquids. We establish the existence of a stable non-Fermi liquid state with a finite-temperature,long-wavelength, isotropic electric conductivity that diverges as a power law in temperature $T$ as $T\to 0$. This two-dimensional system has many properties characteristic of a true isotropic Luttinger liquid, though at zero temperature it becomes anisotropic. This model can easily be extended to three dimensions.
We study finite temperature transport in the Luttinger-Abrikosov-Beneslavskii phase -- an interacting, scale invariant, non-Fermi liquid phase found in quadratic semimetals. We develop a kinetic equation formalism to describe the d.c. transport properties, which are dominated by collisions, and compute the shear viscosity $η$. The ratio of shear viscosity to entropy density $η/s$ is a measure of the strength of interaction between the excitations of a quantum fluid. As a consequence of the quantum critical nature of the system, $η/ s$ is a universal number and we find it to be consistent with …
The "failed Kondo insulator" CeNiSn has long been suspected to be a nodal metal, with a node in the hybridization matrix elements. Here we carry out a series of Nernst effect experiments to delineate whether the severely anisotropic magnetotransport coefficients do indeed derive from a nodal metal or can simply be explained by a highly anisotropic Fermi surface. Our experiments reveal that despite an almost 20-fold anisotropy in the Hall conductivity, the large Nernst signal is isotropic. Taken in conjunction with the magnetotransport anisotropy, these results provide strong support for an iso…
We study systems of coupled spin-gapped and gapless Luttinger liquids. First, we establish the existence of a sliding Luttinger liquid phase for a system of weakly coupled parallel quantum wires, with and without disorder. It is shown that the coupling can {\it stabilize} a Luttinger liquid phase in the presence of disorder. We then extend our analysis to a system of crossed Luttinger liquids and establish the stability of a non-Fermi liquid state: the crossed sliding Luttinger liquid phase (CSLL). In this phase the system exhibits a finite-temperature, long-wavelength, isotropic electric cond…
We analyze a model of two-leg Hubbard ladders weakly coupled by interladder tunneling. At half filling a semimetallic state with small Fermi pockets is induced beyond a threshold tunneling strength. The sign changes in the single electron Green's function relevant for the Luttinger Sum Rule now take place at surfaces with both zeroes and infinities with important consequences for the interpretation of ARPES experiments. Residual interactions between electron and hole-like quasi-particles cause a transition to long range order at low temperatures. The theory can be extended to small doping lead…
We determine the shear viscosity and the dc electrical conductivity of interacting three-dimensional Luttinger semimetals, which have a quadratic band touching point in the energy spectrum, in the hydrodynamic regime. It is well-known that when the chemical potential is right at the band touching point the long-range Coulomb interaction induces the Luttinger-Abrikosov-Beneslavskii (LAB) phase at T = 0, which is an interacting, scale-invariant, non-Fermi-liquid state of electrons. Upon combining the renormalization-group (RG) analysis near the upper critical spatial dimension of four with the B…
Recently, the notion of topological phases of matter has been extended to higher-order incarnations, supporting gapless modes on even lower dimensional boundaries, such as corners and hinges. We here identify a collection of cubic spin-3/2 fermions with biquadratic touching of Kramers degenerate valence and conduction bands as a platform to strain-engineer higher-order topological (HOT) phases: external uniaxial strain gives birth to a HOT Dirac semimetal or an insulator, depending on its sign, featuring topological \emph{hinge} modes in the strain direction. The insulator in fact exhibits \em…
It is shown that an external magnetic field applied to a spin-S isotropic zigzag chain induces a phase with spontaneously broken parity, characterized by long range ordering of vector chirality. This is in contrast to the two-component Luttinger liquid scenario proposed in the literature. Relevance to real materials is discussed.
We present a review of theoretical investigations into the Kohn-Luttinger nonphonon superconductivity mechanism in various 3D and 2D repulsive electron systems described by the Fermi-gas, Hubbard, and Shubin-Vonsovsky models. Phase diagrams of the superconducting state are considered, including regions of anomalous $s$-, $p$-, and $d$-wave pairing. The possibility of a strong increase in the superconducting transition temperature $T_c$ even for a low electron density is demonstrated by analyzing the spin-polarized case or the two-band situation. The Kohn-Luttinger theory explains or predicts s…
We study the competition between Kondo screening and frustrated magnetism on the non-symmorphic Shastry-Sutherland Kondo lattice at a filling of two conduction electrons per unit cell. A previous analysis of this model identified a set of gapless partially Kondo screened phases intermediate between the Kondo-destroyed paramagnet and the heavy Fermi liquid. Based on crystal symmetries, we argue that (i)~both the paramagnet and the heavy Fermi liquid are {\it semimetals} protected by a glide symmetry; and (ii)~partial Kondo screening breaks the symmetry, removing this protection and allowing the…
We show that the problem of impurity tunneling in a Luttinger liquid of electrons with spin is solvable in the spin isotropic case ($g_σ=2$, $g_ρ$ arbitrary). The resulting integrable model is similar to a two channel anisotropic Kondo model, but with the impurity spin in a "cyclic representation" of the quantum algebra $su(2)_q$ associated with the anisotropy. Using exact, non-perturbative techniques we study the RG flow, and compute the DC conductance. As expected from the analysis of Kane and Fisher we find that the IR fixed point corresponds to two separate leads. We also prove an exact du…
Using a recently proposed perturbative numerical renormalization-group algorithm, we explore the connection between quantum criticality and the emergence of Luttinger liquid physics in $t-J$ chains coupled by frustrated interactions. This study is built on an earlier finding that at the maximally frustrated point, the ground state of weakly-coupled Heisenberg chains is disordered, the transverse exchanges being irrelevant. This result is extended here to transverse couplings up to $J_{\perp}=0.6$, and we argue that it may also be valid at the isotropic point. A finite size analysis of coupled …
We study one-dimensional Kondo Lattices (KL) which consist of itinerant electrons interacting with Kondo impurities (KI) - localized quantum magnetic moments. We focus on KL with isotropic exchange interaction between electrons and KI and with a high KI density. The latter determines the principal róle of the indirect interaction between KI for the low energy physics. Namely, the Kondo physics becomes suppressed and all properties are governed by spin ordering. We present a first-ever comprehensive analytical theory of such KL at an arbitrary doping and predict a variety of regimes with differ…
For uniform systems of spin-less fermions in d spatial dimensions with d > 1, interacting through the isotropic two-body potential v(r-r'), a celebrated theorem due to Luttinger (1961) states that under the_assumption_ of validity of the many-body perturbation theory the self-energy Sigma(k;epsilon), with 0 <,= k <,~ 3 k_F (where k_F stands for the Fermi wavenumber), satisfies the following universal asymptotic relation as epsilon approaches the Fermi energy epsilon_F: Im[Sigma(k;epsilon)] ~ -,+ alpha_k (epsilon-epsilon_F)^2, epsilon >,< epsilon_F, with alpha_k >,= 0. As this is, by definition…
We measured the optical reflectivity of [001]-oriented $n$-doped Cd$_{3}$As$_{2}$ in a broad frequency range (50 - 22000 cm$^{-1}$) for temperatures from 10 to 300 K. The optical conductivity, $σ(ω) = σ_{1}(ω) + {\rm i}σ_{2}(ω)$, is isotropic within the (001) plane; its real part follows a power law, $σ_{1}(ω) \propto ω^{1.65}$, in a large interval from 2000 to 8000 cm$^{-1}$. This behavior is caused by interband transitions between two Dirac bands, which are effectively described by a sublinear dispersion relation, $E(k) \propto \lvert k \rvert ^{0.6}$. The momentum-averaged Fermi velocity of…
We uncover the properties of complex tensor (d-wave) superconducting order in three-dimensional Rarita-Schwinger-Weyl semimetals that host pseudospin-3/2 fermions at a fourfold linear band crossing point. Although the general theory of d-wave order was originally developed for materials displaying quadratic band touching, it directly applies to the case of semimetals with linear dispersion, several candidate compounds of which have been discovered experimentally very recently. The spin-3/2 nature of the fermions allows for the formation of spin-2 Cooper pairs which may be described by a comple…
We study transport in three dimensional Weyl semimetals with N isotropic Weyl nodes in the presence of Coulomb interactions or disorder at temperature T. In the interacting clean limit, we determine the conductivity by solving a quantum Boltzmann equation within a `leading log' approximation and find it to be proportional to T, upto logarithmic factors arising from the flow of couplings. In the noninteracting disordered case, we compute the finite-frequency Kubo conductivity and show that it exhibits distinct behaviors for omega << T and omega >> T: in the former regime we recover the results …
We study the thermoelectric properties of a double-Weyl fermion system, possibly realized in $\mathrm{HgCr_2Se_4}$ and $\mathrm{SrSi_2}$, by a semi-classical Boltzmann transport theory. We investigate different relaxation processes including short-range disorder and electron-electron interaction on the thermoelectric transport coefficients. It is found that the anisotropy of the band dispersion for in-plane and out-of-plane momentum directions affects the relaxation time for transport in different directions. The transport also exhibits an interesting directional dependence on the chemical pot…
Magnetic-field-induced Tomonaga-Luttinger liquid and Bose-Einstein condensate phases in an organic S=1 Haldane chain
University of Zagreb（萨格勒布大学）；Université Paul Sabatier（保罗·萨巴蒂尔大学）；Osaka Metropolitan University（大阪公立大学）；Osaka Prefecture University（大阪府立大学）；Laboratoire National des Champs Magnétiques Intenses, LNCMI-CNRS（强磁场国家实验室）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Helmholtz-Zentrum Dresden-Rossendorf（德累斯顿-罗斯多夫亥姆霍兹中心）
Recent progresses in condensed matter physics, such as graphene, topological insulator and Weyl semimetal, often origin from the specific topological symmetries of their lattice structures. Quantum states with different degrees of freedom, e.g. spin, valley, layer, etc., arise from these symmetries, and the coherent superpositions of these states form multiple energy subbands. The pseudospin, a concept analogy to the Dirac spinor matrices, is a successful description of such multi-subband systems. When the electron-electron interaction dominates, many-body quantum phases arise. They usually ha…
The 2-leg t-J ladder forms a spin liquid at half-filling which evolves to a Luther-Emery liquid upon doping. Our aim is to obtain a complete phase diagram for isotropic coupling (i.e. rungs and legs equal) as a function of electron density n and the ratio J/t (>0). Two known limiting cases are: n<1/2 which is a single band Luttinger liquid and small hole doping for J/t close to 0 which is a Nagaoka ferromagnet. Using Lanczos techniques we examine the region between the Nagaoka and Luther-Emery phases for 1>n>1/2. We find evidences for gapless behavior in both spin and charge channels for J/t<0…
We study the spontaneous breaking of rotational symmetry in the helical surface state of three-dimensional topological insulators due to strong electron-electron interactions, focusing on time-reversal invariant nematic order. Owing to the strongly spin-orbit coupled nature of the surface state, the nematic order parameter is linear in the electron momentum and necessarily involves the electron spin, in contrast with spin-degenerate nematic Fermi liquids. For a chemical potential at the Dirac point (zero doping), we find a first-order phase transition at zero temperature between isotropic and …
We study a one-dimensional system that consists of an electron gas coupled to a spin-1/2 chain by Kondo interaction away from half-filling. We show that zero-temperature transitions between phases with "small" and "large" Fermi momenta can be continuous. Such a continuous but Fermi-momentum-changing transition arises in the presence of spin anisotropy, from a Luttinger liquid with a small Fermi momentum to a Kondo-dimer phase with a large Fermi momentum. We have also added a frustrating next-nearest-neighbor interaction in the spin chain to show the possibility of a similar Fermi-momentum-chan…
