Collisionless Transport Close to a Fermionic Quantum Critical Point in Dirac Materials

Quantum transport close to a critical point is a fundamental, but enigmatic problem due to fluctuations, persisting at all length scales. We report the scaling of optical conductivity (OC) in the \emph{collisionless} regime ($\hbar \omega \gg k_B T$) in the vicinity of a relativistic quantum critical point, separating two-dimensional ($d=2$) massless Dirac fermions from a fully gapped insulator or superconductor. Close to such critical point gapless fermionic and bosonic excitations are strongly coupled, leading to a \emph{universal} suppression of the inter-band OC as well as of the Drude peak (while maintaining its delta function profile) inside the critical regime, which we compute to the leading order in $1/N_f$- and $\epsilon$-expansions, where $N_f$ counts fermion flavor number and $\epsilon=3-d$. Correction to the OC at such a non-Gaussian critical point due to the long-range Coulomb interaction and generalizations of these scenarios to a strongly interacting three-dimensional Dirac or Weyl liquid are also presented, which can be tested numerically and possibly from non-pertubative gauge-gravity duality, for example.Comment: Published version in PRL: 5+epsilon Pages, 2 Figures (Supplementary Materials as Ancillary file: 4 pages

Unconventional superconductivity in nearly flat bands in twisted bilayer graphene

Flat electronic bands can accommodate a plethora of interaction driven quantum phases, since kinetic energy is quenched therein and electronic interactions therefore prevail. Twisted bilayer graphene, near so-called the "magic angles", features \emph{slow} Dirac fermions close to the charge-neutrality point that persist up to high-energies. Starting from a continuum model of slow, but strongly interacting Dirac fermions, we show that with increasing chemical doping away from the charge-neutrality point, a time-reversal symmetry breaking, valley pseudo-spin-triplet, topological $p+ip$ superconductor gradually sets in, when the system resides at the brink of an anti-ferromagnetic ordering (due to Hubbard repulsion), in qualitative agreement with recent experimental findings. The $p+ip$ paired state exhibits quantized spin and thermal Hall conductivities, polar Kerr and Faraday rotations. Our conclusions should also be applicable for other correlated two-dimensional Dirac materials.Comment: 5 Pages, 2 Figures: Published Version in PRB (Supplementary Materials: 4 Pages, Ancillary file

Optical conductivity of an interacting Weyl liquid in the collisionless regime

Optical conductivity (OC) can serve as a measure of correlation effects in a wide range of condensed matter systems. We here show that the long-range tail of the Coulomb interaction yields a universal correction to the OC in a three-dimensional Weyl semimetal $\sigma(\Omega)=\sigma_0(\Omega)\left[ 1+\frac{1}{N+1} \right]$, where of $\sigma_0(\Omega)=Ne^2_0 \Omega/(12 h v)$ is the OC in the non-interacting system, with $v$ as the actual (renormalized) Fermi velocity of Weyl quasiparticles at frequency $\Omega$, and $e_0$ is the electron charge in vacuum. Such universal enhancement of OC, which depends only on the number of Weyl nodes near the Fermi level ($N$), is a remarkable consequence of an intriguing conspiracy among the quantum-critical nature of an interacting Weyl liquid, marginal irrelevance of the long-range Coulomb interaction and the violation of hyperscaling in three dimensions, and can directly be measured in recently discovered Weyl as well as Dirac materials. By contrast, a local density-density interaction produces a non-universal correction to the OC, stemming from the non-renormalizable nature of the corresponding interacting field theory.Comment: 21 Pages, 1 Figure: Published Version in PR

The project finance model in the supply of residential and commercial premises

A supply of dwellings greater than the demand, a reduction in the availability of housing loans and increased credit risk, caused, inter alia, by the financial crisis: these are the basic features of today’s residential property and commercial premises markets in Croatia today. Built but unsold housing units have exposed private investors, who have organised the supply of units within the balance sheet of their firms, to significant risk of underinvestment. The materialisation of this risk is most manifested in the impossibility of funding the core business because of loans that they have agreed on for the construction of dwelling units meant for sale on the market. The paper then proposes a model that, if it were applied, could insure investors to a greater extent against the risk of underinvestment. The supply of dwelling units with protected rentals by the local public sector organised in the traditional manner, i.e. according to a model in which the local public sector figures in the role of investor, distributes the burden of development costs onto the future generations as well. However, practice has shown that traditional models inequitably expose future generations to the risk of a reduction in the quality of this kind of public service. From this point of view the proposed model transfers to the future generation not only the costs but also the obligations to secure equal qualities of public service in such a way that the private investor long-term assumes the risk of the availability of public building. The problem in this kind of organisation of the supply of a public service is double taxation via VAT, changes in the law concerning which are accordingly proposed.project financing, housing construction

Higher Order Topological Phases: A General Principle of Construction

We propose a general principle for constructing higher-order topological (HOT) phases. We argue that if a $D$-dimensional first-order or regular topological phase involves $m$ Hermitian matrices that anti-commute with additional $p-1$ mutually anti-commuting matrices, it is conceivable to realize an $n$th-order HOT phase, where $n=1, \cdots, p$, with appropriate combinations of discrete symmetry-breaking Wilsonian masses. An $n$th-order HOT phase accommodates zero modes on a surface with codimension $n$. We exemplify these scenarios for prototypical three-dimensional gapless systems, such as a nodal-loop semimetal possessing SU(2) spin-rotational symmetry, and Dirac semimetals, transforming under (pseudo-)spin-$\frac{1}{2}$ or 1 representations. The former system permits an unprecedented realization of a fourth-order phase, without any surface zero modes. Our construction can be generalized to HOT insulators and superconductors in any dimension and symmetry class.Comment: Published Version in PRB Rapid Comm. (Editors' Suggestion): 5+epsilon Pages and 4 figures (Supplementary Materials: 8 Pages + 9 Figures

Itinerant quantum multi-criticality of two dimensional Dirac fermions

We analyze emergent quantum multi-criticality for strongly interacting, massless Dirac fermions in two spatial dimensions ($d=2$) within the framework of Gross-Neveu-Yukawa models, by considering the competing order parameters that give rise to fully gapped (insulating or superconducting) ground states. We focus only on those competing orders, which can be rotated into each other by generators of an exact or emergent chiral symmetry of massless Dirac fermions, and break $O(S_1)$ and $O(S_2)$ symmetries in the ordered phase. Performing a renormalization group analysis by using the $\epsilon=(3-d)$ expansion scheme, we show that all the coupling constants in the critical hyperplane flow toward a new attractive fixed point, supporting an \emph{enlarged} $O(S_1+S_2)$ chiral symmetry. Such a fixed point acts as an exotic quantum multi-critical point (MCP), governing the \emph{continuous} semimetal-insulator as well as insulator-insulator (for example antiferromagnet to valence bond solid) quantum phase transitions. In comparison with the lower symmetric semimetal-insulator quantum critical points, possessing either $O(S_1)$ or $O(S_2)$ chiral symmetry, the MCP displays enhanced correlation length exponents, and anomalous scaling dimensions for both fermionic and bosonic fields. We discuss the scaling properties of the ratio of bosonic and fermionic masses, and the increased dc resistivity at the MCP. By computing the scaling dimensions of different local fermion bilinears in the particle-hole channel, we establish that most of the four fermion operators or generalized density-density correlation functions display faster power law decays at the MCP compared to the free fermion and lower symmetric itinerant quantum critical points. Possible generalization of this scenario to higher dimensional Dirac fermions is also outlined.Comment: 15 pages, 9 figures; replaced by the version published in Phys. Rev.

Restoration of the magnetic hc/e-periodicity in unconventional superconductors

We consider the energy of the filled quasiparticle's Fermi sea of a macroscopic superconducting ring threaded by an hc/2e-vortex, when the material of the ring is of an unconventional pairing symmetry. The energy relative to the one for the hc/e-vortex configuration is finite, positive, and inversely proportional to ring's inner radius. We argue that the existence of this energy in unconventional superconductors removes the commonly assumed degeneracy between the odd and the even vortices, with the loss of the concomitant hc/2e periodicity in external magnetic field as a consequence. This macroscopic quantum effect should be observable in nanosized unconventional superconductors with a small phase stiffness, such as deeply underdoped YBCO with Tc < 5K.Comment: 4 RevTex pages, 2 figures, published versio

Emergent Lorentz symmetry near fermionic quantum critical points in two and three dimensions

We study the renormalization group flow of the velocities in the field theory describing the coupling of the massless quasi-relativistic fermions to the bosons through the Yukawa coupling, as well as with both bosons and fermions coupled to a fluctuating $U(1)$ gauge field in two and three spatial dimensions. Different versions of this theory describe quantum critical behavior of interacting Dirac fermions in various condensed-matter systems. We perform an analysis using one-loop $\epsilon-$expansion about three spatial dimensions, which is the upper critical dimension in the problem. In two dimensions, we find that velocities of both charged fermions and bosons ultimately flow to the velocity of light, independently of the initial conditions, the number of fermionic and bosonic flavors, and the value of the couplings at the critical point. In three dimensions, due to the analyticity of the gauge field propagator, both the $U(1)$ charge and the velocity of light flow, which leads to a richer behavior than in two dimensions. We show that all three velocities ultimately flow to a common terminal velocity, which is non-universal and different from the original velocity of light. Therefore, emergence of the Lorentz symmetry in the ultimate infrared regime seems to be a rather universal feature of this class of theories in both two and three dimensions.Comment: 19 pages, 4 figures: Published version, added discussion, new references, typos correcte