Microscopic determination of the Coulomb singular domain in gapless semiconductors of the second kind (three-dimensional quadratic band touching): scaling exponents, scaling functions, and conditions for experimental realization

In a gapless semiconductor of the second kind - a crystal such as gray tin, HgTe or HgSe whose conduction and valence bands touch quadratically at one point in momentum space - the density of states vanishes at the touching point, so the electron gas cannot screen the long-range Coulomb interaction at long wavelengths. Abrikosov and Beneslavskii (1971) showed that inside a low-energy singular domain $\omega\ll\omega_0=me^4/\varepsilon_0^2$, $k\ll k_0=me^2/\varepsilon_0$ this interaction is nonperturbative and destroys quasiparticles, and they posited self-similar scaling forms for the interacting Green function and Coulomb vertex with undetermined exponents $\alpha$ and $\nu$ and undetermined scaling functions $g$ and $d$, from which they derived power laws for the conductivity, heat capacity, dielectric function and Hall coefficient. The open problem is the microscopic determination of this interacting ground state at the physical spatial dimension and band degeneracy - the values of $\alpha$, $\nu$, $g$ and $d$, equivalently the fermion anomalous dimension and dynamical exponent of the Luttinger-Abrikosov-Beneslavskii non-Fermi liquid, and whether that scale-invariant state survives at all or gives way to a symmetry-broken (nematic Mott) insulator - together with the exceptional case $\nu=3$, the effect of electron-hole mass asymmetry, and the quantitative impurity-density and band-overlap conditions for observing the predicted power laws in real materials. Fifty years of follow-up work determines the exponents only in uncontrolled-extrapolation limits and disputes the stability of the non-Fermi-liquid state at physical parameters, leaving the exact solution and the experimental realization open.

Background

Gapless semiconductors of the second kind are crystals in which the conduction and valence bands touch at a single point of the Brillouin zone (usually the center, $\mathbf{k}=0$) and, in the one-electron approximation, both disperse quadratically, $\varepsilon(\mathbf{k})\propto k^2$, near the contact. In the older classification, gapless semiconductors of the first kind have linear dispersion at the contact, for which the Coulomb interaction produces only weak logarithmic corrections, whereas the second kind is genuinely strongly coupled at low energy. The classic examples are gray tin ($\alpha$-Sn) and the mercury chalcogenides HgTe and HgSe; the same band structure, including its spin-orbit-driven $j=3/2$ realization described by the Luttinger Hamiltonian (the standard $4\times4$ effective Hamiltonian for a band touching of states with total angular momentum $j=3/2$), appears in modern materials such as the pyrochlore iridate $\mathrm{Pr}_2\mathrm{Ir}_2\mathrm{O}_7$ and the half-Heusler compound YPtBi. The modern name for this low-energy structure is a three-dimensional quadratic-band-touching (QBT) semimetal or Luttinger semimetal.

The reason the second kind is singular is the vanishing density of states: for a quadratic contact in three dimensions $\rho(\varepsilon)\propto\sqrt{\varepsilon}\to0$ as $\varepsilon\to0$, so at charge neutrality (Fermi level at the touching point) there are no carriers to screen the Coulomb potential statically. At the same time the group velocity $v=k/m$ tends to zero with $k$ (here $m$ is the effective mass, $\varepsilon_0$ the background dielectric constant, $e$ the electron charge), so the effective coupling $e^2/(\varepsilon_0 v)$ grows without bound as $k\to0$. Abrikosov and Beneslavskii (1971) identified the resulting low-energy window - the singular domain $\omega\ll\omega_0=me^4/\varepsilon_0^2$, $k\ll k_0=me^2/\varepsilon_0$ - inside which perturbation theory in the interaction fails, and, borrowing the scaling method of strong-coupling field theory, they posited for the exact single-particle Green function (propagator) and the exact screened Coulomb vertex the self-similar forms

$$G(\mathbf{k},\omega)=\omega_0^{-1}\left(\frac{k}{k_0}\right)^{-\alpha}\,g\!\left(\frac{\omega/\omega_0}{(k/k_0)^{\nu}}\right),\qquad \Gamma(\mathbf{k},\omega)=\frac{\omega_0}{k_0^{3}}\left(\frac{k}{k_0}\right)^{\nu-3}\,d\!\left(\frac{\omega/\omega_0}{(k/k_0)^{\nu}}\right),$$

with unknown critical exponents $\alpha$ and $\nu$ and unknown scaling functions $g$ and $d$. In modern renormalization-group (RG) language, $\nu$ is the dynamical exponent $z$ (the scaling of frequency with momentum) and $\alpha=2-\eta_\psi$, where $\eta_\psi$ is the fermion anomalous dimension (the interaction-induced correction to the canonical power-law scaling of the fermion propagator); the corresponding scale-invariant interacting state, with no long-lived quasiparticle excitations - a non-Fermi liquid - is now called the Luttinger-Abrikosov-Beneslavskii (LAB) phase.

From the ansatz alone, the 1971 paper derived parameter-light predictions that remain its experimental fingerprint: the electronic heat capacity $C\sim k_0^3(T/\omega_0)^{3/\nu}$, the dc conductivity $\sigma\sim\varepsilon_0\omega_0(T/\omega_0)^{1/\nu}$, the dielectric function $\varepsilon(\omega)\sim\varepsilon_0(\omega_0/\omega)^{1-1/\nu}$, the Hall coefficient $R\sim(1/eck_0^3)(T/\omega_0)^{-3/\nu}$, an impurity-dominated regime $\sigma_{imp}\sim\varepsilon_0\omega_0(N_i/k_0^3)^{1/3}$ when the impurity density $N_i$ is not negligible, and parameter-free combinations $RC\sim1/(ec)$, $R\sigma^3\sim e^5/(c\hbar^3)$ and $\chi\sigma\sim(e^4/c^2\hbar^3)T$ that are independent of $m$ and $\varepsilon_0$ ($\chi$ is the magnetic susceptibility, $c$ the speed of light, $\hbar$ the reduced Planck constant). Static screening is singular: the effective potential acquires a linear-in-$q$ term, $V(q)\approx4\pi e^2/(\varepsilon^*q^2+me^2qa)$, i.e. a real-space decay $V(r)\propto1/r^2$, a result later rederived and confirmed.

The 1971 paper itself left the following explicitly open: (i) the numerical values of $\alpha$ and $\nu$, a method for computing them, and more generally the scaling functions $g$ and $d$, from a microscopic Hamiltonian (Sec. 7: the self-similar forms are posited as an assumption with only the inequality constraints $\nu>1$, $\alpha<2$, $\alpha<\nu$ derived, and the paper states that the theory "does not give the exact value of the constant, and also of the functions $g$, $q$, and $d$"); (ii) the exceptional case $\nu=3$, at which the zero-energy density of states would remain finite - the value at which the density-of-states exponent $3/\nu-1$ implicit in $C\sim T^{3/\nu}$ (i.e. $\rho\propto\varepsilon^{3/\nu-1}$) vanishes, while the derived constraint $\nu>1$ still holds - left unanalyzed; (iii) the general case of considerably different electron and hole effective masses, treated only qualitatively; and (iv) the experimental conditions: the intrinsic singular domain survives only if the impurity density is low enough relative to the scale $k_0$ (a criterion the 1971 paper left at the order-of-magnitude level, and a purity not achieved in then-existing samples), and the candidate materials HgTe and HgSe may acquire a small spin-orbit-induced band overlap that could destroy the singular domain, with no quantitative criterion given (Abrikosov's 1974 sequel concluded that the singular regions "may be difficult to seek out" even in the purest samples).

The problem was reformulated in the 2010s. The QBT band structure with Coulomb interaction has upper critical dimension $d_c=4$ (above which the interaction is perturbatively irrelevant), so controlled expansions exist in $4-\epsilon$ and $2+\epsilon$ spatial dimensions and at large fermion flavor number $N$; the physical case is $d=3$ with $N=1$, i.e. a single fourfold-degenerate touching as in HgTe and gray tin. Moon, Xu, Kim and Balents (2013) showed at large $N$ that the state is a stable non-Fermi liquid (the LAB phase) protected by time-reversal and cubic symmetries, and computed its universal power-law exponents. Herbut, Janssen and collaborators then found, in several controlled schemes, that below a critical flavor number $N_c(3)\approx1.9$-$2.6$ the LAB fixed point collides with and annihilates against a nematic quantum critical point (a fixed-point collision), so the physical $N=1$ system should instead undergo a transition, at a low but finite temperature $T_c\approx0.1$-$1\,\mathrm{K}$ in clean samples, to a nematic topological Mott insulator - an interaction-driven insulator whose broken symmetry is a quadrupolar (nematic) distortion of the band structure and whose surface states are topological. Whether the LAB non-Fermi liquid or the nematic insulator is the true ground state of the physical $d=3$, $N=1$ system remains disputed in the literature, and the exponents and scaling functions at that point are known only by extrapolation. Abrikosov and Beneslavskii (1971) flagged that "it is quite possible that it may turn out that the interaction leads to instability of the obtained spectrum" and posited the scaling forms as an assumption never established at the physical point.

Problem Statement

From the microscopic Hamiltonian of a clean, charge-neutral, three-dimensional gapless semiconductor of the second kind (quadratic band touching) - the $j=3/2$ Luttinger Hamiltonian coupled by the long-range Coulomb interaction, or an equivalent quadratic-band-touching model - with the physical band degeneracy ($N=1$, a single fourfold-degenerate touching as in gray tin, HgTe) and, separately, with general unequal electron and hole effective masses, determine the complete interacting low-energy theory inside the Abrikosov-Beneslavskii singular domain $\omega\ll\omega_0=me^4/\varepsilon_0^2$, $k\ll k_0=me^2/\varepsilon_0$. Specifically:

(1) Determine the interacting ground state and its universal low-energy structure: whether the scale-invariant non-Fermi-liquid state (Luttinger-Abrikosov-Beneslavskii, LAB) with the Abrikosov-Beneslavskii scaling forms holds at the physical spatial dimension $d=3$ and physical band degeneracy, or whether it is destabilized in favor of a symmetry-broken insulating state; and, for whatever state is realized, compute the exponents $\alpha$ and $\nu$ (equivalently the fermion anomalous dimension and dynamical exponent) and the scaling functions $g$ and $d$ of the Green function and effective Coulomb vertex, together with the resulting observable power laws ($\sigma\propto T^{1/\nu}$, $C\propto T^{3/\nu}$, $\varepsilon(\omega)\propto\omega^{1/\nu-1}$, $R\propto T^{-3/\nu}$, and the parameter-free combinations $RC$, $R\sigma^3$, $\chi\sigma$).

(2) If the exponent $\nu$ determined under (1) takes the exceptional value $\nu=3$ (at which the zero-energy density of states would remain finite), analyze that exceptional case in its own right (the special-case scaling structure and its observable consequences).

(3) Determine how general electron-hole mass asymmetry modifies the singular domain and the exponents.

(4) Establish the quantitative conditions under which the intrinsic singular domain survives in real materials - the maximum impurity density $N_i$ and the maximum band overlap (e.g. the spin-orbit-induced overlap in HgTe and HgSe) compatible with observing the scaling regime - so that the predicted power laws (or their preemption by a symmetry-breaking transition) can be demonstrated experimentally in candidate materials such as $\alpha$-Sn, HgTe, HgSe, pyrochlore iridates or half-Heuslers.

All four sub-questions are in scope. Sub-questions (1), (3) and (4) are mutually independent: resolving any one of them settles that sub-question while leaving the others open. Sub-question (2) is conditional on (1): if the exponent determined under (1) is not $\nu=3$, the exceptional case does not arise and (2) is settled by that statement; if $\nu=3$ is realized, (2) requires the special-case analysis, which the general determination of $\nu$ does not supply. Each sub-question may be reported separately. Within (1), the ground-state determination (scale-invariant LAB non-Fermi liquid versus a symmetry-broken insulating state) and the determination of the exponents $\alpha$, $\nu$ and the scaling functions $g$, $d$ are separate outcomes: either may be established or refuted on its own, and a decision on the realized ground state does not by itself resolve the exponent and scaling-function determination.

References

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