Complete characterization of the pinch-off quantum critical point of a point contact in a quantum spin Hall insulator: boundary conformal field theory, universal scaling, fixed-point classification, and experimental signatures

A quantum point contact connecting the helical edge states of a two-dimensional quantum spin Hall insulator undergoes, for Luttinger parameter $1/2<g<1$, a pinch-off quantum phase transition governed by a nontrivial unstable 'intermediate' boundary fixed point $P$. Teo and Kane (Phys. Rev. B 79, 235321 (2009)) computed the critical conductance and crossover scaling function $\mathcal{G}_g(X)$ only in solvable limits ($g=1-\epsilon$, $g=1/2+\epsilon$, and the self-dual conductance at $g=1/\sqrt{3}$), and left open: (i) an exact analytic or boundary conformal field theory description of $P$ and of $\mathcal{G}_g(X)$ for generic $g$; (ii) numerical computation of these quantities at generic $g$; (iii) a systematic classification of all boundary fixed points and their stability; and (iv) the experimental determination and engineering of $g$, together with observation of the predicted signatures (universal pinch-off scaling, distinct temperature exponents of different conductance components, vanishing skew conductance $G_{XY}\sim T^{\gamma}$, and fractionalized shot noise of the time-reversal-breaking insulating phase for $g<1/2$). The 2020 equivalence of this problem to a two-channel Luttinger-liquid-topological superconductor junction reaffirms (i) as the foremost open problem.

Background

A two-dimensional quantum spin Hall insulator (QSHI, also called a 2D topological insulator) is a bulk band insulator whose boundaries host one-dimensional 'helical' edge channels: for each direction of motion only one spin polarization propagates, and the pairing is protected by time-reversal symmetry. Because the counterpropagating modes are the two spin components of a single Kramers pair (a pair of states related to each other by time reversal and therefore degenerate in zero magnetic field), Coulomb interactions make each edge a helical Luttinger liquid — a one-dimensional correlated state characterized by a Luttinger parameter $g<1$ measuring interaction strength (smaller $g$ means stronger repulsion; $g=1$ is free fermions).

A quantum point contact (QPC) is a short gate-defined constriction that brings the top and bottom edges of a QSHI bar into tunneling contact. Teo and Kane showed that the Luttinger-liquid theory of this four-terminal device maps onto the weak-link problem of a spinful Luttinger liquid (SLL) with charge and spin Luttinger parameters locked by the helical structure to $g_{\rho}=g$ and $g_{\sigma}=1/g$. In this mapping, the gate voltage $V_G$ tunes the point contact through a zero-temperature quantum phase transition — the 'pinch-off' — between a perfectly transmitting charge-conductor/spin-conductor (CC) phase at small $V_G$ and a perfectly reflecting charge-insulator/spin-insulator (II) phase at large $V_G$. For $1/2<g<1$ the transition is governed by a nontrivial unstable 'intermediate' fixed point $P$, a descendant of fixed points previously found in SLL impurity problems. Near $P$, the two-terminal conductance obeys the scaling form $G_0(T,V_G) \sim (2e^2/h)\,\mathcal{G}_g(X)$ with crossover variable $X\propto (V_G-V_G^*)/T^{\alpha_g}$, where $\alpha_g$ is a critical exponent; $\mathcal{G}_g(X)$ is a universal crossover scaling function.

Two theoretical tools give partial access. First, the weak-link problem can be recast as quantum Brownian motion (QBM) — the dissipative motion of a particle on a two-dimensional periodic potential whose lattice geometry depends on $g_{\rho},g_{\sigma}$. In other parameter regimes of the SLL, Yi and Kane mapped such QBM problems onto the multichannel Kondo model (a localized spin coupled to several independent conduction-electron screening channels, whose over-screened critical point is a non-Fermi liquid), and Affleck, Oshikawa and Saleur gave a boundary conformal field theory (BCFT — the description of a two-dimensional conformal field theory on a half-space with conformally invariant boundary conditions, which classify the fixed points of quantum impurity problems) of the triangular-lattice QBM with intermediate-mobility boundary states, related to the boundary three-state Potts model. In the analogous spinless problem, the crossover scaling function was computed exactly for all interaction strengths (Fendley, Ludwig and Saleur, for point contacts in the fractional quantum Hall effect). Second, in the QSHI problem itself, Teo and Kane obtained $\mathcal{G}_g$ and $\alpha_g$ exactly in three solvable limits: $g=1-\epsilon$ (weak interaction, logarithmic renormalization of the scattering matrix), $g=1/2+\epsilon$ (fermionizable, fixed point approaching the mixed charge-insulator/spin-conductor phase), and $g=1/\sqrt{3}$, where the QBM potential has triangular ($C_6$) lattice symmetry and a self-duality argument fixes the critical conductance $G^*=(\sqrt{3}-1)e^2/h$ — but not the full crossover function. Beyond these limits the source proposed a numerical Monte Carlo determination of $\mathcal{G}_g$ and the exponents in the absence of an exact solution, an approach pursued in path-integral Monte Carlo work on the single-impurity spinful Luttinger liquid (Hamamoto, Imura and Kato).

The same paper also predicted, beyond the pinch-off transition itself: a four-terminal conductance matrix whose different components scale with different powers of temperature, including a 'skew' conductance $G_{XY}$ that vanishes as $T^{\gamma}$ with $\gamma\ge 2$ — a direct diagnostic of the helical edge structure; and, for strong interactions $g<1/2$ with spin-nonconserving (time-reversal-preserving) spin-orbit scattering at the junction, a stable time-reversal-symmetry-breaking insulating (TBI) phase in which transport is carried by spinless chargons and chargeless spinons, with characteristic low-frequency shot-noise correlations. An experimental estimate gives $g\sim 0.8$ for HgTe quantum wells, so the exponents (e.g. $\alpha_g\sim 0.02$) are small and reaching the exotic regimes requires engineering smaller $g$, e.g. via the dielectric environment or the bare edge velocity.

In 2020 Kane, Giuliano and Affleck proved that this helical point contact (the 'Teo-Kane' model) is dual to the Affleck-Giuliano model of two single-channel Luttinger liquid leads coupled to a Majorana zero mode at the end of a topological superconductor, related in the free-fermion limit $K=1$ (where $K$ is the Luttinger parameter of the junction leads) by the triality of $SO(8)$ — the exceptional outer automorphism of the $SO(8)$ Lie algebra that cyclically permutes its vector and two spinor representations. This equivalence translates all perturbative results between the two problems and predicts the global behavior of the critical conductance $G^*(K)$; however, both models' intermediate fixed points have been analyzed only in perturbative limits, and a comprehensive BCFT classification remains lacking — a general analysis being obstructed by the fact that for continuous values of the Luttinger parameter the lead conformal field theory is non-rational (it does not decompose into a finite spectrum of primary fields, so the finite rational-CFT classification machinery for boundary states does not apply).

Problem Statement

For a quantum point contact between the helical edges of a quantum spin Hall insulator, modeled as a weak link in a spinful Luttinger liquid with parameters locked to $g_{\rho}=g$, $g_{\sigma}=1/g$, complete the theory of the pinch-off quantum critical point that separates the transmitting (charge-conductor/spin-conductor) and insulating (charge-insulator/spin-insulator) phases for $1/2<g<1$, and test it experimentally. The single overarching objective is a complete, quantitative characterization of this boundary quantum critical point — its fixed-point structure and its universal transport scaling — together with its experimental validation. Specifically, the unresolved components are: (1) an exact analytic characterization of the nontrivial unstable intermediate fixed point $P$ and of its universal crossover scaling function $\mathcal{G}_g(X)$ (and critical exponent $\alpha_g$) for generic interaction parameter $g$; (2) in the absence of an analytic solution, a numerical computation of $\mathcal{G}_g(X)$ and the critical exponents for generic, experimentally relevant $g$; (3) a systematic classification and stability analysis of all fixed points of the problem (seven appear already in the weak-interaction analysis, including multicritical and spin-nonconserving fixed points), with their physical interpretations; and (4) on the experimental side, the determination and engineering of the Luttinger parameter $g$ in quantum spin Hall insulator materials and the observation of the predicted signatures: the universal pinch-off scaling collapse, the distinct temperature exponents of the different four-terminal conductance components, the vanishing skew conductance $G_{XY}\sim T^{\gamma}$ with $\gamma\ge 2$, and, for $g<1/2$, the fractionalized shot-noise signatures of the time-reversal-breaking insulating phase.

References

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