[Paper Review] Soliton Fermi sea in models of Ising-coupled Kondo impurities

Stefan Kehrein, Matthias Vojta|arXiv (Cornell University)|Aug 20, 2002

Quantum and electron transport phenomena2 references4 citations

TL;DR

This paper studies Ising-coupled Kondo impurities, showing that at the Toulouse point the system maps exactly to an Anderson impurity model with a novel solitonic Fermi sea. Using flow equations, the analysis is extended beyond the Toulouse point for weak Ising couplings, revealing a collective Kondo screening of pseudospin degrees of freedom via soliton excitations, a phenomenon absent in SU(2)-symmetric Kondo models.

ABSTRACT

We study a model of Ising-coupled Kondo impurities that can be applied to quantum dots with capacitance coupling, coupled qubits with an incoherent environment, etc. We show that this model becomes equivalent to a Anderson impurity model with a novel solitonic Fermi sea. We derive exact results at the Toulouse point, and use the flow equation method to extend this analysis away from the Toulouse point for not too large Ising couplings.

Motivation & Objective

• To understand the low-energy physics of Kondo impurities coupled via Ising-type $S^z$-$S^z$ interactions, relevant to quantum dots and coupled qubits.
• To address the breakdown of SU(2) symmetry in multi-impurity Kondo systems, where Ising coupling leads to new quantum phases.
• To extend exact results from the Toulouse point to finite Ising couplings using the flow equation method.
• To generalize the mapping to $N$-impurity systems with infinite-range Ising coupling, revealing collective Kondo screening via solitonic excitations.
• To identify conditions under which a quantum phase transition may occur due to the interplay between Kondo screening and ferromagnetic Ising coupling.

Proposed method

• Map the Ising-coupled Kondo model to an effective Anderson impurity model at the Toulouse point using exact transformations.
• Apply the flow equation method (Wegner's method) to systematically derive effective Hamiltonians away from the Toulouse point.
• Use refermionization to express the transformed Hamiltonian in terms of soliton excitations in the Fermi sea.
• Derive a non-constant hybridization function $\Delta_i(\epsilon)$ that captures the crossover between low- and high-energy regimes.
• Establish a criterion for validity: $|K| \ll \min_i T_K^i / |\lambda - 1|$, ensuring perturbative control over the flow equation approximation.
• Generalize the mapping to $N$ impurities with infinite-range $S_i^z S_j^z$ coupling, leading to an $N$-channel Anderson model with density-density interaction.

Experimental results

Research questions

• RQ1 How does Ising coupling between Kondo impurities alter the low-energy fixed point compared to SU(2)-symmetric coupling?
• RQ2 Can the Toulouse point mapping be extended beyond exact solvability to finite Ising couplings using controlled approximations?
• RQ3 What is the role of soliton excitations in the Fermi sea in screening the pseudospin degrees of freedom in Ising-coupled Kondo systems?
• RQ4 Under what conditions does a quantum phase transition occur when the Ising coupling becomes strong relative to the Kondo scale?
• RQ5 How does the collective Kondo scale $T_K^{\text{coll}}$ emerge in $N$-impurity systems with infinite-range Ising coupling?

Key findings

• At the Toulouse point, the Ising-coupled Kondo model maps exactly to an Anderson impurity model with a Fermi sea composed of fermionic soliton excitations.
• The soliton Fermi sea enables collective screening of the two-fold degenerate ground state of the isolated double-impurity system below a collective Kondo scale $T_K^{\text{coll}}$.
• For weak Ising couplings satisfying $|K| \ll \min_i T_K^i / |\lambda - 1|$, the flow equation method yields reliable approximations, extending the Toulouse point results.
• The hybridization function $\Delta_i(\epsilon)$ shows a power-law dependence $\sim |\epsilon|^{\lambda_i^2 - 1}$ at high energies, with $\Delta_i(0) \sim T_K^i$.
• For large positive $K$, the effective interaction becomes ferromagnetic, potentially destroying Kondo screening and leading to a doubly degenerate ground state.
• In the $N$-impurity case with infinite-range Ising coupling, the system maps to an $N$-channel Anderson model with a density-density interaction, supporting a collective Kondo scale $T_K^{\text{coll}}$.