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arXiv:1603.00031 · Physical review. B./Physical review. B

Superconducting quantum criticality in three-dimensional Luttinger semimetals

Igor Boettcher, Igor F. Herbut

Published May 23, 2016• 82 citations
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Abstract

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 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 emergent rotational and particle-hole symmetries. Some features of the criticality, such as oscillatory corrections to scaling and its enhanced symmetry, are discussed.

DOI
10.1103/physrevb.93.205138

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We study the quantum electrodynamics of Luttinger fermions with quadratic band-crossing dispersion in three dimensions. The model can be viewed as the low-energy effective theory of a putative U(1) quantum spin liquid with fermionic Luttinger spinons, or as an extension of the Luttinger-Abrikosov-Beneslavskii (LAB) model that accounts for transverse gauge fluctuations with finite photon velocity. Aided by a renormalization group analysis below four dimensions, we elucidate the presence and stability of quantum critical phenomena in this model. We find that the non-Fermi liquid LAB phase is stable against gauge fluctuations and can thus also be viewed as a U(1) spin liquid with gapless Luttinger spinons. We discover a multicritical point with Lifshitz scaling that corresponds to a time-reversal symmetry-breaking quantum phase transition from the LAB state to a chiral spin liquid with spinon Landau levels and birefringent emergent photons. This multicritical point is characterized by a finite fermion-photon coupling in the infrared and can be viewed as a fermionic analog of the Rokhsar-Kivelson point in three-dimensional quantum dimer models.
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We study a model for a quantum critical point in two spatial dimensions between a semimetallic phase, characterized by a stable quadratic Fermi node, and an ordered phase, in which the spectrum develops a band gap. The quantum critical behavior can be computed exactly, and we explicitly derive the scaling laws of various observables. While the order-parameter correlation function at criticality satisfies the usual power law with anomalous exponent ${\ensuremath{\eta}}_{\ensuremath{\phi}}=2$, the correlation length and the expectation value of the order parameter exhibit essential singularities upon approaching the quantum critical point from the insulating side, akin to the Berezinskii-Kosterlitz-Thouless transition. The susceptibility, on the other hand, has a power-law divergence with non-mean-field exponent $\ensuremath{\gamma}=2$. On the semimetallic side, the correlation length remains infinite, leading to an emergent scale invariance throughout this phase.
We investigate a semimetal-superconductor phase transition of two-dimensional Dirac electrons at zero temperature by large-scale and essentially unbiased quantum Monte Carlo simulations for the half-filled attractive Hubbard model on the triangular lattice, in the presence of alternating magnetic $\ensuremath{\pi}$ flux, that is introduced to construct two Dirac points in the one-particle bands at the Fermi level. This phase transition is expected to describe quantum criticality of the chiral XY class in the framework of the Gross-Neveu model, where, in the ordered phase, the U(1)U(1)U(1) symmetry is spontaneously broken and a mass gap opens in the excitation spectrum. We compute the order parameter of the sss-wave superconductivity and estimate the quasiparticle weight from the long-distance behavior of the single-particle Green's function. These calculations allow us to obtain the critical exponents of this transition in a reliable and accurate way. Our estimate for the critical exponents is in good agreement with those obtained for a transition to a Kekul\'e valence bond solid, where an emergent U(1)U(1)U(1) symmetry is proposed [Z.-X. Li et al., Nat. Commun. 8, 314 (2017)].
We analyze the quantum phase transition between a semimetal and a superfluid in a model of attractively interacting fermions with a linear dispersion. The quantum critical properties of this model cannot be treated by the Hertz-Millis approach since integrating out the fermions leads to a singular Landau-Ginzburg order parameter functional. We therefore derive and solve coupled renormalization-group equations for the fermionic degrees of freedom and the bosonic order parameter fluctuations. In two spatial dimensions, fermions and bosons acquire anomalous scaling dimensions at the quantum critical point, associated with non-Fermi-liquid behavior and non-Gaussian order parameter fluctuations.
First proposed in the context of elementary particle physics, supersymmetry is an exotic symmetry that relates bosons and fermions but has not yet been observed in nature. While the traditional expectation is that supersymmetry would be first observed in a high-energy physics experiment, an alternative possibility stressed recently is that supersymmetry could be realized as an emergent symmetry at long wavelengths and low energies in interacting many-particle systems. More specifically, it was suggested that supersymmetry emerges at the quantum phase transition between the gapless Dirac semimetal and a gapped superconductor on the surface of a 3D topological insulator. The authors of this paper provide additional support for the conjecture by performing a three-loop renormalization group calculation and showing that the supersymmetric fixed point in 2+1 dimensions remains stable to that order in perturbation theory.