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arXiv:2402.09510 · Physical review. B./Physical review. B

Dissipation driven phase transition in the non-Hermitian Kondo model

Pradip Kattel, Abay Zhakenov, Parameshwar R. Pasnoori, P. Azaria, Natan Andrei

Published May 9, 2025• 17 citations
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Abstract

Non-Hermitian Hamiltonians, as effective models, capture phenomena such as energy dissipation and nonunitary evolution in open quantum systems. New phases and phenomena appear that are not present in their Hermitian counterparts. Such a Hamiltonian, the non-Hermitian Kondo model, has been used to describe inelastic scattering between mobile and confined atoms in an optical lattice [M. Nakagawa, N. Kawakami, and M. Ueda, Phys. Rev. Lett. 121, 203001 (2018)]. Using a combination of Bethe ansatz and perturbative calculation, the authors argued that this model has two distinct phases: the Kondo and the non-Kondo phases, where the impurity is screened and unscreened, respectively. We show, however, that a novel phase termed $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{YSR}$ emerges between the Kondo and unscreened phases. Characterized by two RG invariants: a generalized Kondo temperature (${T}_{K}$) and a loss strength parameter ($\ensuremath{\alpha}$), the system exhibits three distinct phases. In the increasing order of losses, they are: the Kondo phase ($0<\ensuremath{\alpha}<\ensuremath{\pi}/2$), the $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{YSR}$ phase ($\ensuremath{\pi}/2<\ensuremath{\alpha}<3\ensuremath{\pi}/2$), and the local moment phase ($\ensuremath{\alpha}>3\ensuremath{\pi}/2$). Notably, phase transition driven by dissipation occurs across $\ensuremath{\alpha}=\ensuremath{\pi}/2$, where both energetics and different timescales associated with loss play roles.

DOI
10.1103/physrevb.111.l201106

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Non-Hermitian Hamiltonians, as effective models, capture phenomena such as energy dissipation and nonunitary evolution in open quantum systems. New phases and phenomena appear that are not present in their Hermitian counterparts. Such a Hamiltonian, the non-Hermitian Kondo model, has been used to describe inelastic scattering between mobile and confined atoms in an optical lattice [M. Nakagawa, N. Kawakami, and M. Ueda, Phys. Rev. Lett. 121, 203001 (2018)]. Using a combination of Bethe ansatz and perturbative calculation, the authors argued that this model has two distinct phases: the Kondo and the non-Kondo phases, where the impurity is screened and unscreened, respectively. We show, however, that a novel phase termed YSR\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{YSR}YSR\ifmmode~\else˜\fi emerges between the Kondo and unscreened phases. Characterized by two RG invariants: a generalized Kondo temperature (TK{T}_{K}TK​) and a loss strength parameter (\ensuremathα\ensuremath{\alpha}\ensuremathα), the system exhibits three distinct phases. In the increasing order of losses, they are: the Kondo phase (0<\ensuremathα<\ensuremathπ/20<\ensuremath{\alpha}<\ensuremath{\pi}/20<\ensuremathα<\ensuremathπ/2), the YSR\ifmmode~\else˜\fi\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{YSR}YSR\ifmmode~\else˜\fi phase (\ensuremathπ/2<\ensuremathα<3\ensuremathπ/2\ensuremath{\pi}/2<\ensuremath{\alpha}<3\ensuremath{\pi}/2\ensuremathπ/2<\ensuremathα<3\ensuremathπ/2), and the local moment phase (\ensuremathα>3\ensuremathπ/2\ensuremath{\alpha}>3\ensuremath{\pi}/2\ensuremathα>3\ensuremathπ/2). Notably, phase transition driven by dissipation occurs across \ensuremathα=\ensuremathπ/2\ensuremath{\alpha}=\ensuremath{\pi}/2\ensuremathα=\ensuremathπ/2, where both energetics and different timescales associated with loss play roles.
The authors construct here an exactly solvable \ensuremath{\mathcal{P}}\ensuremath{\mathcal{T}}-symmetric non-Hermitian Heisenberg spin chain with complex-conjugate boundary Kondo couplings. They show that the model exhibits three distinct boundary phases: dissipative Kondo screening with unbroken \ensuremath{\mathcal{P}}\ensuremath{\mathcal{T}} symmetry, bound-mode phases featuring spontaneously broken \ensuremath{\mathcal{P}}\ensuremath{\mathcal{T}} symmetry and complex-conjugate eigenvalues, and an unscreened local-moment phase with a purely real spectrum. Benchmarking with non-Hermitian DMRG demonstrates quantitative agreement with Bethe ansatz, validating tensor-network methods for open quantum systems.
Non-Hermitian (NH) Hamiltonians describe open quantum systems, nonequilibrium dynamics, and dissipative processes. Although a rich range of single-particle NH physics has been uncovered, many-body phenomena in strongly correlated NH systems have been far less well studied. The Kondo effect, an important paradigm for strong correlation physics, has recently been considered in the NH setting. Here, we develop an NH generalization of the numerical renormalization group and use it to solve the NH Kondo model. Our nonperturbative solution applies beyond weak coupling, and we uncover a nontrivial phase diagram. The method is showcased by application to the NH pseudogap Kondo model, which we show supports a completely novel phase with a genuine NH stable fixed point and complex eigenspectrum. Our NH numerical renormalization group code, which can be used in regimes and for models inaccessible to, e.g., perturbative scaling and Bethe ansatz, is provided open source.
We analyze the competition between strong correlations and dissipation in quantum impurity systems from the Kondo regime to the valence fluctuation regime by developing a slave-boson theory for a non-Hermitian Anderson impurity model with one-body loss. Notably, in the non-Hermitian Kondo regime, strong correlations qualitatively change the nature of dissipation through renormalization effects, where the effective one-body loss is suppressed and emergent many-body dissipation characterized by the complex-valued hybridization is generated. We unveil the mechanism of a dissipative quantum phase transition of the Kondo state on the basis of this renormalization effect, which counterintuitively enhances the lifetime of the impurity against loss. We also find a crossover from the non-Hermitian Kondo regime to the valence fluctuation regime dominated by one-body dissipation. Our results can be tested in a wide variety of setups such as quantum dots coupled to electronic leads and quantum point contacts in ultracold Fermi gases.
Quantum phase transitions out of equilibrium are outstanding emergent subjects in condensed matter physics with great fundamental importance and challenges. We theoretically investigate here the nonequilibrium quantum phase transition in a generic nano-setup: the pseudogap Kondo model where a Kondo quantum dot couples to two---left (L) and right (R)---voltage-biased fermionic leads with power-law density of states (DOS) with respect to their Fermi levels ${\ensuremath{\mu}}_{L/R}$, ${\ensuremath{\rho}}_{c,L(R)}(\ensuremath{\omega})\ensuremath{\propto}{|\ensuremath{\omega}\ensuremath{-}{\ensuremath{\mu}}_{L(R)}|}^{r}$ with 0<r<10<r<10<r<1. In equilibrium (${\ensuremath{\mu}}_{L}\ensuremath{-}{\ensuremath{\mu}}_{R}=0$) and for 0<r<1/20<r<1/20<r<1/2, with increasing Kondo correlations this model exhibits a quantum phase transition from a unscreened local moment (LM) phase to the Kondo screened phase. At finite bias voltages and near criticality, we discover new nonequilibrium universal scaling behaviors in conductance, conduction electron TTT matrix, and local spin susceptibility via a controlled frequency-dependent renormalization group (RG) approach. The current-induced decoherence is key to understanding these distinct universal nonequilibrium quantum critical regimes. The relevance of our results to experiment is discussed.
Zero temperature phase transitions not only occur in the bulk of quantum systems, but also at boundaries or impurities. We review recent work on quantum phase transitions in impurity models that are generalizations of the standard Kondo model describing the interaction of a localized magnetic moment with a metallic fermionic host. Whereas in the standard case the moment is screened for any antiferromagnetic Kondo coupling as T to 0, the common feature of all systems considered here is that Kondo screening is suppressed due to the competition with other processes. This competition can generate unstable fixed points associated with phase transitions, where the impurity properties undergo qualitative changes. In particular, we discuss the coupling to both non-trivial fermionic and bosonic baths as well as two-impurity models, and make connections to recent experiments.
We use the framework setup recently to compute nonperturbatively inelastic scattering from quantum impurities [G. Zar\'and et al., Phys. Rev. Lett. 93, 107204 (2004)] to study the energy dependence of the single-particle SSS matrix and the inelastic scattering cross section for a number of quantum impurity models. We study the case of the spin S=1∕2S=1∕2S=1∕2 two-channel Kondo model, the Anderson model, and the usual S=1∕2S=1∕2S=1∕2 single-channel Kondo model. We discuss the difference between non-Fermi-liquid and Fermi-liquid models and study how a crossover between the non-Fermi-liquid and Fermi-liquid regimes appears in the case of channel anisotropy for the S=1∕2S=1∕2S=1∕2 two-channel Kondo model. We show that for the most elementary non-Fermi-liquid system, the two-channel Kondo model, half of the scattering remains inelastic even at the Fermi energy. Details of the derivation of the reduction formulas and a simple path integral approach to connect the TTT matrix to local correlation functions are also presented.