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arXiv:2210.04937 · Physical Review Letters

Z3 Parafermion in the Double Charge Kondo Model

D. B. Karki, E. Boulat, Winston Pouse, David Goldhaber‐Gordon, Andrew K. Mitchell, Christophe Mora

Published Apr 6, 2023• 27 citations
arXivHTMLPDF

Abstract

Quantum impurity models with frustrated Kondo interactions can support quantum critical points with fractionalized excitations. Recent experiments [W. Pouse et al., Nat. Phys. (2023)NPAHAX1745-247310.1038/s41567-022-01905-4] on a circuit containing two coupled metal-semiconductor islands exhibit transport signatures of such a critical point. Here, we show using bosonization that the double charge-Kondo model describing the device can be mapped in the Toulouse limit to a sine-Gordon model. Its Bethe-ansatz solution shows that a Z_{3} parafermion emerges at the critical point, characterized by a fractional 1/2ln(3) residual entropy, and scattering fractional charges e/3. We also present full numerical renormalization group calculations for the model and show that the predicted behavior of conductance is consistent with experimental results.

DOI
10.1103/physrevlett.130.146201

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Quantum impurity models with frustrated Kondo interactions can support quantum critical points with fractionalized excitations. Recent experiments [W. Pouse et al., Nat. Phys. (2023)NPAHAX1745-247310.1038/s41567-022-01905-4] on a circuit containing two coupled metal-semiconductor islands exhibit transport signatures of such a critical point. Here, we show using bosonization that the double charge-Kondo model describing the device can be mapped in the Toulouse limit to a sine-Gordon model. Its Bethe-ansatz solution shows that a Z_{3} parafermion emerges at the critical point, characterized by a fractional 1/2ln(3) residual entropy, and scattering fractional charges e/3. We also present full numerical renormalization group calculations for the model and show that the predicted behavior of conductance is consistent with experimental results.
We study the influence of many-body interactions on the transport properties in a two-site charge Kondo circuit recently implemented in a hybrid metal-semiconductor double-quantum dot device [W. Pouse {\it et al.}, Nat. Phys. {\bf 19}, 492 (2023)]. There emerge two principal types of interactions: (i) an intrinsic one, described by the Luttinger liquid model, and (ii) an induced one, which appears due to the coupling of the system to an Ohmic environment. Case (i) could be achieved if the charge Kondo circuit operates in the fractional quantum Hall regime, while case (ii) can be implemented via a finite number of open ballistic channels coupled to both the quantum dots. We demonstrate that the conductance scaling for the case of strong and weak interdot coupling is fully determined by the effective interaction parameter, which is the combination of the fractional filling factor ν=1/mν=1/mν=1/m and the number of transmitting channels. Furthermore, we predict that the fractional filling factor ννν defines a universal Kondo scaling in the vicinity of a special triple quantum critical point featured by the emergence of a Z3\mathbb{Z}_3Z3​ parafermion.
We study the influence of many-body interactions on the transport properties in a two-site charge Kondo circuit recently implemented in a hybrid metal-semiconductor double-quantum dot device [W. Pouse et al., Nat. Phys. 19, 492 (2023)]. There emerge two principal types of interactions: (i) an intrinsic one, described by the Luttinger liquid model, and (ii) an induced one, which appears due to the coupling of the system to an Ohmic environment. Case (i) could be achieved if the charge Kondo circuit operates in the fractional quantum Hall regime, while case (ii) can be implemented via a finite number of open ballistic channels coupled to both the quantum dots. We demonstrate that the conductance scaling for the case of strong and weak interdot coupling is fully determined by the effective interaction parameter, which is the combination of the fractional filling factor $\ensuremath{\nu}=1/m$ and the number of transmitting channels. Furthermore, we predict that the fractional filling factor $\ensuremath{\nu}$ defines a universal Kondo scaling in the vicinity of a special triple-quantum critical point featured by the emergence of a Z3{\mathbb{Z}}_{3}Z3​ parafermion.
Quantum impurity models provide a central framework for correlated electron physics, with quantum dots enabling controlled experimental realizations. While their weak-coupling behavior is well understood through mappings to Kondo Hamiltonians, the opposite regime of highly transparent contacts has lacked a controlled theoretical description. Using the functional renormalization group (FRG), we resolve this regime for the three-channel charge Kondo device of Iftikhar et al. [Tunable quantum criticality and super-ballistic transport in a "charge" Kondo circuit, Science 360, 1315 (2018)SCIEAS0036-807510.1126/science.aan5592], benchmarking against conformal field theory by reproducing the universal zero-frequency conductance and, crucially, going beyond it to obtain the full frequency crossover of the conductance and the full temperature crossover of the impurity entropy, together with a continuous line of fixed points for interacting leads. These results establish FRG as a powerful nonperturbative tool for quantum impurity problems in regimes inaccessible to conventional approaches, with direct implications for mesoscopic experiments.
Hybrid metal-semiconductor quantum circuits offer a controlled setting for realizing and probing strongly correlated quantum matter. A largely open question that may be addressed by such simulators is whether coupled critical states can combine into new non-Fermi liquid (NFL) phases. Inspired by the recent realization of a two-site charge-Kondo circuit [W. Pouse \textit{et al.}, \href{https://www.nature.com/articles/s41567-022-01905-4}{Nat. Phys. \textbf{19}, 492 (2023)}], we uncover the fate of coupled Kondo anyons emerging from the two-channel-Kondo critical point at each site. Bosonization and quantum Brownian motion methods yield the full quasiballistic phase diagram and low-temperature transport properties, while numerical renormalization group calculations resolve the complementary weak-coupling limit. Together, these provide a unified description, and reveal robust NFL phases with continuously tunable transport and susceptibility exponents. Stronger inter-site coupling ultimately produces a screened Fermi liquid.
Unconventional quantum states defying the ubiquitous Fermi-liquid paradigm can emerge in the presence of strong electronic correlations. Among these, non-Abelian anyons - such as Majorana zero modes and Fibonacci anyons - are of particular interest for topological quantum computing due to their non-integer quantum dimensions d>1, which allows for protected non-local encoding and processing of quantum information. However, despite considerable efforts, the unambiguous characterisation of such anyons via transport measurements has proved challenging. Instead, here we provide experimental evidence for the low-temperature fractional entropy Delta S associated with a single anyon, which directly implies its non-Abelian character through the relation Delta S = kB ln(d). This thermodynamic signature is measured in metal-semiconductor quantum circuits engineered to realize quantum-critical states from frustrated interactions. Using a micrometre-scale metallic island coupled to two or three electronic leads, we tune the system to two-channel and three-channel Kondo critical points. By measuring the island charge and exploiting a thermodynamic Maxwell relation, we estimate the entropy associated with the anyons that emerge in these critical states. Our observations reveal fractional values, exposing non-Abelian anyons. The corresponding scaling dimensions are consistent with theoretical predictions for a Majorana zero mode Delta S = kB ln(sqrt(2)) and a Fibonacci anyon Delta S = kB ln(1 +sqrt(5))/2 for two and three channels. These findings establish entropy measurements as a powerful tool for characterizing exotic quantum states.
The phenomenon of charge fractionalization describes the emergence of novel excitations with fractional quantum numbers, as predicted in strongly correlated systems such as spin liquids. We elucidate that precisely such an unusual effect may occur in the simplest possible non-Fermi liquid, the two-channel Kondo effect. To bring this concept down to experimental test, we study nonequilibrium transport through a device realizing the charge two-channel Kondo critical point in a recent experiment by Iftikhar et al. [Nature (London) 526, 233 (2015)NATUAS0028-083610.1038/nature15384]. The shot noise at low voltages is predicted to result in a universal Fano factor e^{*}/e=1/2. This allows us to experimentally identify elementary transport processes of emergent fermions carrying half-integer charge.