Frustrated Kondo impurity triangle: A simple model of deconfinement

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Frustrated Kondo impurity triangle: A simple model of deconfinement

Elio J. König, Piers Coleman, and Yashar Komijani

Phys. Rev. B 104, 115103 – Published 3 September 2021

Abstract

The concepts of deconfinement and topological order are of great current interest for quantum information science and for our understanding of quantum materials. Here, we introduce a simple model of three antiferromagnetically coupled Kondo impurities, a Kondo triangle, which can be used to further extend the application of these concepts to electronic systems. We show that, by tuning the magnetic frustration, the Kondo triangle undergoes a quantum phase transition between two phases of unbroken symmetry, signaling a phase transition beyond the Landau paradigm. We demonstrate that the frustrated spin liquid phase is described by a three-channel Kondo (3CK) fixed point and thus displays an irrational ground state degeneracy. Using an Abrikosov pseudofermion representation, this quantum state is categorized by an emergent $U$ (1) gauge field and its projective symmetry group. The gauge theory is deconfining in the sense that a miniature Wilson loop orders and topological defects (instantons in the gauge field) are expelled. This phase persists in the presence of moderate Kondo screening until proliferation of topological defects leads to a quantum phase transition to an unfrustrated Fermi liquid phase. Based on this evidence, we propose that the 3CK phase displays topological order in a similar sense as gapless spin liquids.

Received 4 December 2020
Revised 25 June 2021
Accepted 16 July 2021

DOI:https://doi.org/10.1103/PhysRevB.104.115103

Physics Subject Headings (PhySH)

Anyons Fractionalization Kondo effect Quantum phase transitions Quantum spin liquid

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Elio J. König ^1,2,*, Piers Coleman^1,3, and Yashar Komijani^1,4

¹Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, New Jersey 08854, USA
²Max Planck Institute for Solid State Research, Heisenbergstraße 1, 70569 Stuttgart, Germany
³Department of Physics, Royal Holloway, University of London, Egham, Surrey TW20 0EX, United Kingdom
⁴Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221-0011, USA

^*elio.j.koenig@gmail.com

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Vol. 104, Iss. 11 — 15 September 2021

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Images

Figure 1
(a) An antiferromagnetic triangle, where each spin is coupled to its own conduction bath, caricatures a spin-liquid competing with a Fermi liquid (FL). (b) When $T_{K} / J_{H}$ is large, each spin is individually Kondo screened [local FL (LFL), right inset]. In contrast, at the smallest $T_{K} / J_{H}$ , the ground state manifold of the impurity forms an effective spin (left inset), and the system develops a three-channel Kondo (3CK) phase, in which instantons of the emergent gauge theory are irrelevant. Analogously to confinement in two- plus one-dimensional quantum electrodynamics ( ${QED}_{3}$ ), instantons proliferate beyond a critical $T_{K} / J_{H}$ (red star) and restore an ordinary (L)FL.
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Figure 2
(a) Pictorial representation of the mean-field Hamiltonian. (b) Schematic mean-field phase diagram. (c) Zero temperature, mean-field behavior of $t$ , $Δ = π ρ V^{2}$ as a function of the Doniach parameter (here, $J_{4} = 0.3 J_{H}$ , $J_{s} \sim 0.29 J_{H}$ ). The position where $t_{c}$ , defined in Eq. (21) below, crosses $t$ defines the confinement-deconfinement quantum phase transition at which phase slips proliferate (red star; here, $N = 4$ and $q = \frac{1}{4}$ ).
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Figure 3
(a) Ground state energies at $T_{K} = 0$ as a function of filling $q$ , comparing rotationally symmetric solutions (labeled by their flux $Φ$ ) with symmetry broken states with one (two) nonzero $t_{m}$ [graphically labeled by triangles with one (two) thick bonds]. (b) Single particle spectra of spinons at $T_{K} = 0$ with helicity quantum numbers $h \in {0, \pm 2 π / 3}$ for different flux configurations $Φ = 0, \pm 2 π$ . Instantons map $Φ \to Φ \pm 2 π$ and thereby reshuffle the wave functions but leave the spectrum unaltered.
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Figure 4
Graphical illustration of mean-field solutions. (a)–(c) Each point on the black curves corresponds to a solution of Eq. (8c) for a given value of the Doniach parameter $π T_{K} / J_{H}$ . For $q < \frac{1}{2}$ ( $q > \frac{1}{2}$ ), we concentrate on $d < 0$ ( $d > 0$ ) (other solutions are “false vacua” indicated as thin lines). For comparison, we include analogous curves of solutions in the limit $T_{K} = 0$ [ $t > 0, V = 0$ (red) and $t < 0, V = 0$ (blue)] as well as $J_{H} = 0$ [ $t = 0, V > 0$ (green)]. The vertical gray dashed lines indicate the position of $q = \frac{1}{4}, \frac{1}{3}$ . For $q = \frac{1}{4}$ , there are up to two nontrivial solutions with $- π < d < 0$ (orange and pink circles, denoted $d_{<}$ and $d_{>}$ , respectively), while for $q = \frac{1}{3}$ , there is only one nontrivial solution (pink circle) in addition to the solution $d = - π$ corresponding to complete Kondo breakdown ( $V = 0$ ). The corresponding mean-field energy is plotted with the same color code in (d) and (e) and compared with the solutions where either $V = 0$ or $t = 0$ .
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