We study non-equilibrium dynamics and quantum quenches in a recent gauge/gravity duality model for a strongly coupled system interacting with a magnetic impurity with SU(N ) spin. At large N , it is convenient to write the impurity spin as a bilinear in Abrikosov fermions. The model describes an RG flow triggered by the marginally relevant Kondo operator. There is a phase transition at a critical temperature, below which an operator condenses which involves both an electron and an Abrikosov fermion field. This corresponds to a holographic superconductor in AdS2 and models the impurity screening. We quench the Kondo coupling either by a Gaussian pulse or by a hyperbolic tangent, the latter taking the system from the condensed to the uncondensed phase or vice-versa. We study the time dependence of the condensate induced by this quench. The timescale for equilibration is generically given by the leading quasinormal mode of the dual gravity model. This mode also governs the formation of the screening cloud, which is obtained as the decrease of impurity degrees of freedom with time. In the condensed phase, the leading quasinormal mode is imaginary and the relaxation of the condensate is over-damped. For quenches whose final state is close to the critical point of the large N phase transition, we study the critical slowing down and obtain the combination of critical exponents zν = 1. When the final state is exactly at the phase transition, we find that the exponential ringing of the quasinormal modes is replaced by a power-law behaviour of the form ∼ t −a sin(b log t). This indicates the emergence of a discrete scale invariance.
We study non-equilibrium dynamics and quantum quenches in a recent gauge/gravity duality model for a strongly coupled system interacting with a magnetic impurity with SU(N ) spin. At large N , it is convenient to write the impurity spin as a bilinear in Abrikosov fermions. The model describes an RG flow triggered by the marginally relevant Kondo operator. There is a phase transition at a critical temperature, below which an operator condenses which involves both an electron and an Abrikosov fermion field. This corresponds to a holographic superconductor in AdS2 and models the impurity screening. We quench the Kondo coupling either by a Gaussian pulse or by a hyperbolic tangent, the latter taking the system from the condensed to the uncondensed phase or vice-versa. We study the time dependence of the condensate induced by this quench. The timescale for equilibration is generically given by the leading quasinormal mode of the dual gravity model. This mode also governs the formation of the screening cloud, which is obtained as the decrease of impurity degrees of freedom with time. In the condensed phase, the leading quasinormal mode is imaginary and the relaxation of the condensate is over-damped. For quenches whose final state is close to the critical point of the large N phase transition, we study the critical slowing down and obtain the combination of critical exponents zν = 1. When the final state is exactly at the phase transition, we find that the exponential ringing of the quasinormal modes is replaced by a power-law behaviour of the form ∼ t −a sin(b log t). This indicates the emergence of a discrete scale invariance.
We propose a model of a strongly-interacting two-impurity Kondo system based on the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence, also known as holography. In a Landau Fermi Liquid, the single-impurity Kondo effect is the screening of an impurity spin at low temperature T . The two-impurity Kondo model then describes the competition between the Kondo interaction and the Heisenberg interaction between two impurity spins, also called the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction. For spin-1/2 impurities, that competition leads to a quantum phase transition from a Kondo-screened phase to a phase in which the two impurity spins screen one another. Our holographic model is based on a (1 + 1)-dimensional CFT description of the two-impurity Kondo model, reliable for two impurities with negligible separation in space. We consider only impurity spins in a totally anti-symmetric representation of an SU(N ) spin symmetry. We employ a large-N limit, in which both Kondo and RKKY couplings are double-trace, and both Kondo and inter-impurity screening appear as condensation of single-trace operators at the impurities’ location. We perform the holographic renormalization of our model, which allows us to identify the Kondo and RKKY couplings as boundary conditions on fields in AdS. We numerically compute the phase diagram of our model in the plane of RKKY coupling versus T , finding evidence for a quantum phase transition from a trivial phase, with neither Kondo nor inter-impurity screening, to a non-trivial phase, with both Kondo and anti-ferromagnetic inter-impurity screening. More generally we show, just using SU(N ) representation theory, that ferromagnetic correlations must be absent at leading order in the large-N limit. Our holographic model may be useful for studying many open problems involving strongly-interacting quantum impurities, including for example the Kondo lattice, relevant for describing the heavy fermion compounds.
We calculate entanglement and impurity entropies in a recent holographic model of a magnetic impurity interacting with a strongly coupled system. There is an RG flow to an IR fixed point where the impurity is screened, leading to a decrease in impurity degrees of freedom. This information loss corresponds to a volume decrease in our dual gravity model, which consists of a codimension one hypersurface embedded in a BTZ black hole background in three dimensions. There are matter fields defined on this hypersurface which are dual to Kondo field theory operators. In the large N limit, the formation of the Kondo cloud corresponds to the condensation of a scalar field. The entropy is calculated according to the Ryu‐Takayanagi prescription. This requires to determine the backreaction of the hypersurface on the BTZ geometry, which is achieved by solving the Israel junction conditions. We find that the larger the scalar condensate gets, the more the volume of constant time slices in the bulk is reduced, shortening the bulk geodesics and reducing the impurity entropy. This provides a new non‐trivial example of an RG flow satisfying the g‐theorem. Moreover, we find explicit expressions for the impurity entropy which are in agreement with previous field theory results for free electrons. This demonstrates the universality of perturbing about an IR fixed point.
We calculate entanglement and impurity entropies in a recent holographic model of a magnetic impurity interacting with a strongly coupled system. There is an RG flow to an IR fixed point where the impurity is screened, leading to a decrease in impurity degrees of freedom. This information loss corresponds to a volume decrease in our dual gravity model, which consists of a codimension one hypersurface embedded in a BTZ black hole background in three dimensions. There are matter fields defined on this hypersurface which are dual to Kondo field theory operators. In the large N limit, the formation of the Kondo cloud corresponds to the condensation of a scalar field. The entropy is calculated according to the Ryu-Takayanagi prescription. This requires to determine the backreaction of the hypersurface on the BTZ geometry, which is achieved by solving the Israel junction conditions. We find that the larger the scalar condensate gets, the more the volume of constant time slices in the bulk is reduced, shortening the bulk geodesics and reducing the impurity entropy. This provides a new non-trivial example of an RG flow satisfying the g-theorem. Moreover, we find explicit expressions for the impurity entropy which are in agreement with previous field theory results for free electrons. This demonstrates the universality of perturbing about an IR fixed point.
Prethermalization and transient dynamics of the Multi-Channel Kondo systems under generic quantum quenches: Insights form Large-N Schwinger-Keldysh approach
Understanding out-of-equilibrium many-body quantum systems is crucial in contemporary physics. However, capturing the universal dynamics of such systems remains challenging despite advanced numerical methods. We investigate the multi-channel Kondo impurity (MCKI) model, hosting an over-screened Kondo state with non-Fermi liquid characteristics. Using the large-N Schwinger-Keldysh approach, we study transient dynamics and long-time quasi-equilibrium properties following a sudden change in the Kondo coupling. We consider two initial states: over-screened Kondo and high-temperature Fermi liquid. In the over-screened Kondo state, we observe oscillations in spin-spin correlations and the Kondo order parameter, representing quantum revival of the entangled state. In the high-temperature Fermi liquid state, the absence of oscillations is attributed to the de-phasing mechanism. The system reaches quasi-equilibrium, manifested as incoherent thermalization between the impurity and conduction electrons. We observe a non-vanishing effective temperature difference between the impurity spin (Abrikosov fermion) and the composite boson formed with conduction electrons at the impurity site. This quasi-equilibrium is prethermalization, while complete thermalization occurs on an O(N) timescale with 1/N correction. We discuss the quantum cooling effect and quantum Boltzmann equations. Our study establishes a foundation for investigating large-N quantum field theory treatment of quantum many-body systems, revealing universal properties and a fresh perspective on prethermalization.
Abstract We investigate the boundary phenomena that arise in a finite-size XX spin chain interacting through an XX interaction with a spin − 1 2 impurity located at its edge. Upon Jordan–Wigner transformation, the model is described by a quadratic Fermionic Hamiltonian. Our work displays, within this ostensibly simple model, the emergence of the Kondo effect, a quintessential hallmark of strongly correlated physics. We also show how the Kondo cloud shrinks and turns into a single particle bound state as the impurity coupling increases beyond a critical value. In more detail, using both Bethe Ansatz and exact diagonalization techniques, we show that the local moment of the impurity is screened by different mechanisms depending on the ratio of the boundary and bulk coupling J imp J . When the ratio falls below the critical value 2 , the impurity is screened via the Kondo effect. However, when the ratio between the coupling exceeds the critical value 2 an exponentially localized bound mode is formed at the impurity site which screens the spin of the impurity in the ground state. We show that the boundary phase transition is reflected in local ground state properties by calculating the spinon density of states, the magnetization at the impurity site in the presence of a global magnetic field, and the finite temperature susceptibility of the impurity. We find that the spinon density of states in the Kondo phase has the characteristic Lorentzian peak that moves from the Fermi level to the maximum energy of the spinon as the impurity coupling is increased and becomes a localized bound mode in the bound mode phase. Moreover, the impurity magnetization and the finite temperature impurity susceptibility behave differently in the two phases. When the boundary coupling J imp exceeds the critical value 2 J , the model is no longer boundary conformal invariant as a massive bound mode appears at the impurity site.
We revisit the physics of a Kondo impurity coupled to a fermionic host with a diverging power-law density of states near the Fermi level, $\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\sim}{|\ensuremath{\omega}|}^{r}$, with exponent $\ensuremath{-}1<r<0$. Using the analytical understanding of several fixed points, based partially on powerful mappings between models with bath exponents r and $(\ensuremath{-}r)$, combined with accurate numerical renormalization group calculations, we determine thermodynamic quantities within the stable phases and also near the various quantum phase transitions. Antiferromagnetic Kondo coupling leads to strong screening with a negative zero-temperature impurity entropy, while ferromagnetic Kondo coupling can induce a stable fractional spin moment. We formulate the quantum field theories for all critical fixed points of the problem, which are fermionic in nature and allow for a perturbative renormalization-group treatment.