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arXiv:1503.04242 · Physical Review B

Nematic quantum criticality in three-dimensional Fermi system with quadratic band touching

Lukas Janssen, Igor F. Herbut

Published Jul 20, 2015• 83 citations
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Abstract

We construct and discuss the field theory for tensorial nematic order parameter coupled to gapless four-component fermions at the quadratic band touching point in three (spatial) dimensions. Within a properly formulated epsilon-expansion this theory is found to have a quantum critical point, which describes the (presumably continuous) transition from the semimetal into a (nematic) Mott insulator. The latter phase breaks the rotational, but not the time-reversal, symmetry and may be relevant to materials such as gray tin or mercury telluride at low temperatures. The critical point represents a simple quantum analog of the familiar classical isotropic-to-nematic transition in liquid crystals. The properties and the consequences of this quantum critical point are discussed. Its existence supports the scenario of the ``fixed-point collision,'' according to which three-dimensional Fermi systems with quadratic band touching and long-range Coulomb interactions are unstable towards the gapped nematic ground state at low temperatures.

DOI
10.1103/physrevb.92.045117

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We construct and discuss the field theory for tensorial nematic order parameter coupled to gapless four-component fermions at the quadratic band touching point in three (spatial) dimensions. Within a properly formulated epsilon-expansion this theory is found to have a quantum critical point, which describes the (presumably continuous) transition from the semimetal into a (nematic) Mott insulator. The latter phase breaks the rotational, but not the time-reversal, symmetry and may be relevant to materials such as gray tin or mercury telluride at low temperatures. The critical point represents a simple quantum analog of the familiar classical isotropic-to-nematic transition in liquid crystals. The properties and the consequences of this quantum critical point are discussed. Its existence supports the scenario of the ``fixed-point collision,'' according to which three-dimensional Fermi systems with quadratic band touching and long-range Coulomb interactions are unstable towards the gapped nematic ground state at low temperatures.
We argue that a three-dimensional electronic system with the Fermi level at the quadratic band touching point such as HgTe could be unstable with respect to the spontaneous formation of the (topological) Mott insulator at arbitrary weak long-range Coulomb interaction. The mechanism of the instability can be understood as the collision of Abrikosov's non-Fermi liquid fixed point with another, quantum critical, fixed point, which approaches it in the coupling space as the system's dimensionality $d\ensuremath{\rightarrow}{d}_{\text{low}}+$, with the ``lower critical dimension'' 2<dlow<42<{d}_{\text{low}}<42<dlow​<4. Arguments for the existence of the quantum critical point based on considerations in the large-NNN limit in d=3d=3d=3, as well as close to d=2d=2d=2, are given. In the one-loop calculation we find that dlow=3.26{d}_{\text{low}}=3.26dlow​=3.26, and thus above, but not far from three dimensions. This translates into a temperature or energy window (Tc{T}_{\mathrm{c}}Tc​, T∗{T}_{*}T∗​) over which the non-Fermi liquid scaling should still be observable, before the Mott transition finally takes place at the critical temperature ${T}_{\mathrm{c}}\ensuremath{\sim}{T}_{*}\mathrm{exp}[\ensuremath{-}zC/({d}_{\text{low}}\ensuremath{-}d{)}^{1/2}]$. We estimate $C=\ensuremath{\pi}/1.1$, dynamical critical exponent $z\ensuremath{\approx}1.8$, and the temperature scale ${k}_{B}{T}_{*}\ensuremath{\approx}(4m/{m}_{\text{el}}{ϵ}^{2})13.6\text{ }\text{ }\mathrm{eV}$, with mmm as the band mass and ϵϵϵ as the dielectric constant.
At low temperatures, isotropic two-dimensional Fermi systems with quadratic band touching are unstable with respect to a symmetry-broken ground state when interactions are taken into account. Here, the authors demonstrate that in a setup with threefold symmetry only - as realized, for instance, in bilayer graphene - interactions induce a splitting of the quadratic band touching into four Dirac cones. This shifts the instability to a finite interaction strength and leads to a quantum critical point with emergent Lorentz invariance.
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 sss-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<21<z<21<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.
Quadratic band touching (QBT) points are widely observed in two- and three-dimensional (2D and 3D) materials, including bilayer graphene and Luttinger semimetals, and attract significant attention from theory to experiment. However, even in its simplest form, the 2D checkerboard lattice QBT model, the phase diagram characterized by temperature and interaction strength, still remains unknown beyond the weak-coupling regime. Intense debates persist regarding the existence of various interaction-driven insulating states in this system. To address these uncertainties, we employ thermal tensor network simulations, specifically exponential tensor renormalization group and tangent space tensor renormalization group, along with density matrix renormalization group calculations to provide a comprehensive finite-temperature phase diagram for this model and shed light on previous ambiguities. Notably, our findings reveal the emergence of a robust bond-nematic Dirac semimetal phase with distinct thermodynamic properties that set it part from the nematic insulating state and other symmetry-broken states. This previously overlooked feature is found to be ubiquitous in interacting QBT systems. We also discuss the implications of these results for experimental systems such as bilayer graphene and iridate compounds.
We investigate the stability of a quadratic band-crossing point (QBCP) in 2D fermionic systems. At the noninteracting level, we show that a QBCP exists and is topologically stable for a Berry flux +/-2pi if the point symmetry group has either fourfold or sixfold rotational symmetries. This putative topologically stable free-fermion QBCP is marginally unstable to arbitrarily weak short-range repulsive interactions. We consider both spinless and spin-1/2 fermions. Four possible ordered states result: a quantum anomalous Hall phase, a quantum spin Hall phase, a nematic phase, and a nematic-spin-nematic phase.
We present a renormalization-group theory for the onset of Ising-nematic order in a Fermi liquid in two spatial dimensions. This is a quantum phase transition, driven by electron interactions, which spontaneously reduces the point-group symmetry from square to rectangular. The critical point is described by an infinite set of 2+12+12+1-dimensional local field theories, labeled by points on the Fermi surface. Each field theory contains a real scalar field representing the Ising order parameter and fermionic fields representing a time-reversed pair of patches on the Fermi surface. We demonstrate that the field theories obey compatibility constraints required by our redundant representation of the underlying degrees of freedom. Scaling forms for the response functions are proposed and supported by computations up to three loops. Extensions of our results to other transitions of two-dimensional Fermi liquids with broken point-group and/or time-reversal symmetry are noted. Our results extend also to the problem of a Fermi surface coupled to a U(1) gauge field.