[Paper Review] Universal Symmetry-Protected Resonances in a Spinful Luttinger Liquid

Yichen Hu, C. L. Kane|arXiv (Cornell University)|Apr 28, 2016

Quantum and electron transport phenomena3 citations

TL;DR

This paper studies resonant tunneling in a spinful Luttinger liquid, demonstrating universal symmetry-protected resonances governed by a non-Fermi liquid fixed point. At specific interaction strengths (gρ=1/3, gσ=1), the system maps to an SU(3) Kondo problem, enabling exact calculation of on-resonance conductance and temperature-dependent line-shape scaling via boundary conformal field theory.

ABSTRACT

We study the problem of resonant tunneling through a quantum dot in a spinful Luttinger liquid. For a range of repulsive interactions, we find that for symmetric barriers there exist resonances with a universal peak conductance $2g^* e^2/h$ that are controlled by a non-trivial intermediate fixed point. This fixed point is also a quantum critical point separating symmetry-protected topological phases. By tuning the system through resonance, all SPT phases can be accessed. For a particular interaction strength with Luttinger parameters $g_ρ=1/3$ and $g_σ=1$, we show that the problem is equivalent to a two channel $SU(3)$ Kondo problem($SU(3)_2$ CFT). At the Toulouse limit, both problems can be mapped to a quantum Brownian motion model on a Kagome lattice, which in turn is related to the quantum Brownian motion on a honeycomb lattice and the three-channel $SU(2)$ Kondo problem($SU(2)_3$ CFT). "Level-rank duality" in the quantum Brownian motion model relating $SU(2)_k$ CFT to $SU(k)_2$ CFT is also explored. Utilizing the boundary conformal field theory, the on-resonance conductance of our resonant tunneling problem is calculated as well as the scaling dimension of the leading relevant operator. This allows us to compute the scaling behavior of the resonance line-shape as a function of temperature.

Motivation & Objective

• To understand resonant tunneling in a spinful Luttinger liquid with repulsive interactions.
• To identify symmetry-protected topological phases and their quantum critical points in the presence of inversion and time-reversal symmetry.
• To establish a mapping between resonant tunneling and multichannel Kondo problems at the Toulouse limit.
• To calculate the on-resonance conductance and scaling behavior of the resonance line-shape using boundary conformal field theory.
• To explore level-rank duality between SU(k)₂ and SU(2)ₖ conformal field theories via quantum Brownian motion models.

Proposed method

• Utilizes boundary conformal field theory to analyze the resonant tunneling problem in a spinful Luttinger liquid with Luttinger parameters gρ and gσ.
• Maps the resonant tunneling problem to a quantum Brownian motion model on a Kagome lattice at the Toulouse limit.
• Establishes equivalence between the quantum Brownian motion model and a two-channel SU(3) Kondo problem with SU(3)₂ conformal field theory.
• Applies the known solution of the multichannel Kondo problem to compute the on-resonance conductance and scaling dimension of the leading relevant operator.
• Uses renormalization group flow equations to analyze the intermediate fixed point and derive the temperature-dependent resonance line-shape.
• Explores level-rank duality by comparing quantum Brownian motion on honeycomb and Kagome lattices, showing equivalence between SU(2)ₖ and SU(k)₂ CFTs.

Experimental results

Research questions

• RQ1 What are the symmetry-protected topological phases in a spinful Luttinger liquid, and how are they characterized?
• RQ2 How does resonant tunneling in a spinful Luttinger liquid lead to a universal peak conductance, and what controls this behavior?
• RQ3 What is the nature of the intermediate fixed point governing the resonance, and how does it relate to non-Fermi liquid physics?
• RQ4 How is the quantum Brownian motion model on a Kagome lattice related to the SU(3) Kondo problem at the Toulouse limit?
• RQ5 What is the significance of level-rank duality in connecting SU(2)ₖ and SU(k)₂ conformal field theories in this context?

Key findings

• For gρ=1/3 and gσ=1, the resonant tunneling problem maps exactly to a two-channel SU(3) Kondo problem described by SU(3)₂ conformal field theory.
• The on-resonance conductance is universally given by 2g*e²/h, with g* determined by the fixed point, and is independent of microscopic details.
• The scaling dimension of the leading relevant operator is computed, enabling exact derivation of the temperature-dependent resonance line-shape.
• The intermediate fixed point is a quantum critical point separating topologically distinct insulating phases, with non-Fermi liquid behavior.
• Level-rank duality is established via quantum Brownian motion, showing equivalence between SU(2)ₖ and SU(k)₂ CFTs for the Kondo and resonant tunneling problems.
• The system accesses all symmetry-protected topological phases by tuning through the resonance, including a charge-insulating but spin-conducting phase for gρ<1/2.