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Physical review. B./Physical review. B

Prethermalization and transient dynamics of multichannel Kondo systems under generic quantum quenches: Insights from large- N Schwinger-Keldysh approach

Iksu Jang, Po-Yao Chang

Published Apr 18, 2024• 3 citations
Publisher

Abstract

Understanding out-of-equilibrium many-body quantum systems is an essential task in contemporary physics. While advanced numerical methods have been developed, capturing the universal dynamics of generic many-body quantum systems still remains a significant challenge. In this study, we focus on the multichannel Kondo impurity (MCKI) model, an intriguing theoretical model hosting an overscreened Kondo state with non-Fermi-liquid characteristics, which serves as a theoretical platform for investigating universal properties in many-body quantum dynamics. Utilizing the large-$N$ Schwinger-Keldysh approach, we systematically investigate both transient dynamics and long-time quasiequilibrium properties in the MCKI model following a sudden change of the Kondo coupling. Our investigations encompass two distinct initial states: the overscreened Kondo state and the high-temperature Fermi-liquid state. For the overscreened Kondo state initial condition, we observe oscillations in various physical observables, including spin-spin correlations and the Kondo order parameter. These oscillations signify the quantum revival of the entangled state that characterizes the overscreened Kondo state. On the other hand, in the case of the high-temperature Fermi-liquid initial state, the absence of oscillations can be attributed to the dephasing mechanism. Furthermore, we discover that the system reaches a quasiequilibrium on an $O(1)$ timescale. This quasiequilibrium manifests as incoherent thermalization between the impurity and the conduction electrons, in which we observe the nonvanishing effective temperature difference between the Abrikosov fermion representing the impurity spin and the composite boson formed by the Abrikosov fermion and the conduction electrons at the impurity site. This quasiequilibrium can be interpreted as prethermalization. Incorporating the $1/N$ correction, we demonstrate that the system attains complete thermalization on an $O(N)$ timescale. Additionally, we discuss the quantum cooling effect and quantum Boltzmann equations. Our comprehensive study establishes a foundation for investigating quantum many-body systems using large-$N$ quantum field-theory treatment, while our findings reveal several universal properties of quantum dynamics and provide a different perspective on prethermalization.

DOI
10.1103/physrevb.109.144305

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Physical review. B./Physical review. B•Feb 14, 2022•8 citations
Understanding out-of-equilibrium many-body quantum systems is an essential task in contemporary physics. While advanced numerical methods have been developed, capturing the universal dynamics of generic many-body quantum systems still remains a significant challenge. In this study, we focus on the multichannel Kondo impurity (MCKI) model, an intriguing theoretical model hosting an overscreened Kondo state with non-Fermi-liquid characteristics, which serves as a theoretical platform for investigating universal properties in many-body quantum dynamics. Utilizing the large-NNN Schwinger-Keldysh approach, we systematically investigate both transient dynamics and long-time quasiequilibrium properties in the MCKI model following a sudden change of the Kondo coupling. Our investigations encompass two distinct initial states: the overscreened Kondo state and the high-temperature Fermi-liquid state. For the overscreened Kondo state initial condition, we observe oscillations in various physical observables, including spin-spin correlations and the Kondo order parameter. These oscillations signify the quantum revival of the entangled state that characterizes the overscreened Kondo state. On the other hand, in the case of the high-temperature Fermi-liquid initial state, the absence of oscillations can be attributed to the dephasing mechanism. Furthermore, we discover that the system reaches a quasiequilibrium on an O(1)O(1)O(1) timescale. This quasiequilibrium manifests as incoherent thermalization between the impurity and the conduction electrons, in which we observe the nonvanishing effective temperature difference between the Abrikosov fermion representing the impurity spin and the composite boson formed by the Abrikosov fermion and the conduction electrons at the impurity site. This quasiequilibrium can be interpreted as prethermalization. Incorporating the 1/N1/N1/N correction, we demonstrate that the system attains complete thermalization on an O(N)O(N)O(N) timescale. Additionally, we discuss the quantum cooling effect and quantum Boltzmann equations. Our comprehensive study establishes a foundation for investigating quantum many-body systems using large-NNN quantum field-theory treatment, while our findings reveal several universal properties of quantum dynamics and provide a different perspective on prethermalization.
Prethermalization and transient dynamics of the Multi-Channel Kondo systems under generic quantum quenches: Insights form Large-NNN 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.
Many-body systems that saturate the quantum bound on chaos are attracting interest across a wide range of fields. Notable examples include the Sachdev-Ye-Kitaev model and its variations, all characterized by some form or randomness and all-to-all couplings. Here, we study many-body quantum chaos in a quantum impurity model showing non-Fermi-liquid physics, the overscreened multichannel S U ( N ) Kondo model. We exactly compute the low-temperature behavior of the out-of-time order correlator in the limit of large N and large number of channels, K , at a fixed ratio γ = K / N . Because of strong correlations at the impurity site, the spin fractionalizes in auxiliary fermions and bosons. We show that all the degrees of freedom of our theory acquire a Lyapunov exponent that is linear in temperature as T → 0 , with a prefactor that depends on γ . Remarkably, for N = K , the impurity spin displays maximal chaos, while bosons and fermions only reach half of the maximal Lyapunov exponent. Our results highlight two key features: a nondisordered model that is maximally chaotic due to strong correlations at its boundary, with the maximal chaos appearing in a composite gauge-invariant operator, the impurity spin, and not in the auxiliary single-particle degrees of freedom.
In the multichannel Kondo effect, overscreening of a magnetic impurity by conduction electrons leads to a frustrated exotic ground state. It has been proposed that multichannel topological Kondo (MCTK) model involving topological Cooper pair boxes with M M Majorana modes [SO( M M ) “spin”] and N N spinless electron channels exhibits an exotic intermediate coupling fixed point. This intermediate fixed point has been analyzed through large- N N perturbative calculations, which gives a zero-temperature conductance decaying as 1/N^2 1 / N 2 in the large- N N limit. However, the conductance at this intermediate fixed point has not been calculated for generic N N . Using representation theory, we verify the existence of this intermediate-coupling fixed point and find the strong-coupling effective Hamiltonian for the case M=4 M = 4 . Using conformal field theory techniques for SO( M M ), we generalize the notion of overscreening and conclude that the MCTK model is an overscreened Kondo model. We find the fixed-point finite-size energy spectrum and the leading irrelevant operator (LIO). We express the fixed-point conductance in terms of the modular S-matrix of SO( M M ) for general N N , confirming the previous large- N N result. We describe the finite-temperature corrections to the conductance by the LIO and find that they are qualitatively different for the cases N=1 N = 1 and N≥2 N ≥ 2 due to the different fusion outcomes with the current operator. We also compare the multichannel topological Kondo model to the topological symplectic Kondo model.
The traditional multichannel Kondo effect takes place when several gapless metallic electronic channels interact with a localized spin-SSS impurity, with the number of channels nnn exceeding the size of the impurity spin, n>2Sn>2Sn>2S, leading to the emergence of non-Fermi liquid impurity behavior at low temperatures. Here, we show that the effect can be realized even when the electronic degrees of freedom are strongly correlated and gapped. The system under consideration consists of a single spin-12\frac{1}{2}21​ impurity coupled isotropically to nnn spin singlet superconducting channels realized by one-dimensional leads with quasi-long-range superconducting order. The competition between the Kondo and superconducting fluctuations induces multiple distinct ground states and boundary phases depending on the relative strengths of the bulk and boundary interactions. Using the Bethe Ansatz technique, we identify four regimes: an overscreened Kondo phase, a zero-mode phase, a Yu-Shiba-Rusinov (YSR) phase, and a local-moment phase with an unscreened impurity, each with its own experimental characteristic. We describe the renormalization-group flow, the excitation spectrum, and the full impurity thermodynamics in each phase. Remarkably, even in the presence of a bulk mass gap, the boundary critical behavior in the Kondo phase is governed by the same exponents as in the gapless theory with the low-energy impurity sector flowing to the SU(2)nSU(2)_nSU(2)n​ Wess-Zumino-Witten (WZW) fixed point, and the impurity entropy monotonically decreasing as a function of temperature. In both the overscreened Kondo and zero-mode phases, the residual impurity entropy is Simp(T→0)=ln⁡[2cos⁡(π/(n+2))]S_{\mathrm{imp}}(T \to 0) = \ln[2\cos(π/(n+2))]Simp​(T→0)=ln[2cos(π/(n+2))]. In the YSR and unscreened phases on the other hand the impurity entropy exhibits non-monotonic temperature dependence and is effectively free at low temperatures with Simp(T→0)=ln⁡2S_{\mathrm{imp}}(T \to 0) = \ln 2Simp​(T→0)=ln2.
We study how the non-Fermi-liquid nature of the overscreened multichannel Kondo impurity model affects the response to a BCS pairing term that, in the absence of the impurity, opens a gap $\mathrm{\ensuremath{\Delta}}$. We find that the low-energy spectrum in the limit $\mathrm{\ensuremath{\Delta}}\ensuremath{\rightarrow}0$ actually does not correspond to the spectrum strictly at $\mathrm{\ensuremath{\Delta}}=0$. In particular, in the two-channel Kondo model, the $\mathrm{\ensuremath{\Delta}}\ensuremath{\rightarrow}0$ ground state is an orbitally degenerate spin singlet, while it is an orbital singlet with a residual spin degeneracy at $\mathrm{\ensuremath{\Delta}}=0$. In addition, there are fractionalized spin-1/2 subgap excitations whose energy in units of $\mathrm{\ensuremath{\Delta}}$ tends toward a finite and universal value when $\mathrm{\ensuremath{\Delta}}\ensuremath{\rightarrow}0$, as if the universality of the anomalous power-law exponents that characterize the overscreened Kondo effect turned into universal energy ratios when the scale invariance is broken by $\mathrm{\ensuremath{\Delta}}\ensuremath{\ne}0$. This intriguing phenomenon can be explained by the renormalization flow toward the overscreened fixed point and the gap cutting off the orthogonality catastrophe singularities. We also find other non-Fermi-liquid features at finite $\mathrm{\ensuremath{\Delta}}$: the local density of states lacks coherence peaks, the states in the continuum above the gap are unconventional, and the boundary entropy is a nonmonotonic function of temperature. The persistent subgap excitations are characteristic of the non-Fermi-liquid fixed point of the model, and thus depend on the impurity spin and the number of screening channels.
We analyze the single-channel Kondo model using the recently developed unitary renormalization-group (URG) method and obtain a comprehensive understanding of the Kondo screening cloud. The fixed-point low-energy Hamiltonian enables the computation of a plethora of thermodynamic quantities (specific heat, susceptibility, Wilson ratio, etc.) as well as spectral functions, all of which are found to be in good agreement with known results. By integrating out the impurity, we obtain an effective Hamiltonian for the excitations of the electrons comprising the Kondo cloud. This is found to contain both kkk-space number diagonal (Fermi liquid) and off-diagonal four-fermion scattering terms. Our conclusions are reinforced by a URG study of the two-particle entanglement and many-body correlations among members of the Kondo cloud and impurity. The entanglement between the impurity and a cloud electron, as well as between any two cloud electrons, is found to increase under flow towards the singlet ground state at the strong-coupling fixed point. Both the number diagonal and off-diagonal correlations within the conduction cloud are also found to increase as the impurity is screened under the flow, and the latter are found to be responsible for the macroscopic entanglement of the Kondo-singlet ground state. The URG flow enables an analytic computation of the phase shifts suffered by the conduction electrons at the strong-coupling fixed point. This reveals an orthogonality catastrophe between the local-moment and strong-coupling ground states and is related to a change in the Luttinger volume of the conduction bath under the crossover to strong coupling. Our results offer fresh insight into the nature of the emergent many-particle entanglement within the Kondo cloud and pave the way for further investigations in more exotic contexts such as the fixed point of the overscreened multichannel Kondo problem.