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arXiv:1804.06082

Long Time Behvaior of the Kondo Model After a Quench

Eldad Bettelheim

Published Apr 17, 2018• 0 citations
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Abstract

We find the statistical weight of excitations at long times following a quench in the Kondo problem. The weights computed are directly related to the overlap between initial and final states that are, respectively, states close to the Kondo ground state and states close to the normal metal ground state. The overlap is computed making use of the Slavnov approach, whereby a functional representation method is adopted, in order to obtain definite expressions.

DOI
10.48550/arxiv.1804.06082

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We find the statistical weight of excitations at long times following a quench in the Kondo problem. The weights computed are directly related to the overlap between initial and final states that are, respectively, states close to the Kondo ground state and states close to the normal metal ground state. The overlap is computed making use of the Slavnov approach, whereby a functional representation method is adopted, in order to obtain definite expressions.
We give integral equations for the generating function of the cummulants of the work done in a quench for the Kondo model in the thermodynamic limit. Our approach is based on an extension of the thermodynamic Bethe ansatz to non-equilibrium situations. This extension is made possible by use of a large N expansion of the overlap between Bethe states. In particular, we make use of the Slavnov determinant formula for such overlaps, passing to a function-space representation of the Slavnov matrix. We leave the analysis of the resulting integral equations to future work.
We consider a quantum quench from the strongly correlated ground state of the Kondo model to a Fermi sea. We calculate the overlap between the ground states before and after the quench, as well as the Loschmidt echo, i.e., the transition amplitude between the initial state and the evolved state at a time ttt after the quench. The overlap is known to determine the dynamics of the echo at large times. We show, in addition, that the overlap depends algebraically on the emergent Kondo length, with a power-law exponent that is the difference of long and short-time contributions that appear in the echo. Our result suggests that, in general, there may be more information contained in the overlap than previously recognized.
It is well known that the ground states of a Fermi liquid with and without a single Kondo impurity have an overlap that decays as a power law of the system size, expressing the Anderson orthogonality catastrophe. Ground states with two different values of the Kondo couplings have, however, a finite overlap in the thermodynamic limit. This overlap, which plays an important role in quantum quenches for impurity systems, is a universal function of the ratio of the corresponding Kondo temperatures, which is not accessible using perturbation theory or the Bethe ansatz. Using a strategy based on the integrable structure of the corresponding quantum field theory, we propose an exact formula for this overlap, which we check against extensive density matrix renormalization group calculations.
We investigate the time evolution of the Kondo resonance in response to a quench by applying the time-dependent numerical renormalization group (TDNRG) approach to the Anderson impurity model in the strong correlation limit. For this purpose, we derive within the TDNRG approach a numerically tractable expression for the retarded two-time nonequilibrium Green function $G(t+{t}^{\ensuremath{'}},t)$, and its associated time-dependent spectral function, $A(\ensuremath{\omega},t)$, for times ttt both before and after the quench. Quenches from both mixed valence and Kondo correlated initial states to Kondo correlated final states are considered. For both cases, we find that the Kondo resonance in the zero temperature spectral function, a preformed version of which is evident at very short times $t\ensuremath{\rightarrow}{0}^{+}$, only fully develops at very long times $t\ensuremath{\gtrsim}1/{T}_{K}$, where TK{T}_{K}TK​ is the Kondo temperature of the final state. In contrast, the final state satellite peaks develop on a fast time scale $1/\mathrm{\ensuremath{\Gamma}}$ during the time interval $\ensuremath{-}1/\mathrm{\ensuremath{\Gamma}}\ensuremath{\lesssim}t\ensuremath{\lesssim}+1/\mathrm{\ensuremath{\Gamma}}$, where $\mathrm{\ensuremath{\Gamma}}$ is the hybridization strength. Initial and final state spectral functions are recovered in the limits $t\ensuremath{\rightarrow}\ensuremath{-}\ensuremath{\infty}$ and $t\ensuremath{\rightarrow}+\ensuremath{\infty}$, respectively. Our formulation of two-time nonequilibrium Green functions within the TDNRG approach provides a first step towards using this method as an impurity solver within nonequilibrium dynamical mean field theory.
Computing the non-equilibrium dynamics that follows a quantum quench is difficult, even in exactly solvable models. Results are often predicated on the ability to compute overlaps between the initial state and eigenstates of the Hamiltonian that governs time evolution. Except for a handful of known cases, it is generically not possible to find these overlaps analytically. Here we develop a numerical approach to preferentially generate the states with high overlaps for a quantum quench starting from the ground state or an excited state of an initial Hamiltonian. We use these preferentially generated states, in combination with a "high overlap states truncation scheme" and a modification of the numerical renormalization group, to compute non-equilibrium dynamics following a quench in the Lieb-Liniger model. The method is non-perturbative, works for reasonable numbers of particles, and applies to both continuum and lattice systems. It can also be easily extended to more complicated scenarios, including those with integrability breaking.
Computing the non-equilibrium dynamics that follows a quantum quench is difficult, even in exactly solvable models. Results are often predicated on the ability to compute overlaps between the initial state and eigenstates of the Hamiltonian that governs time evolution. Except for a handful of known cases, it is generically not possible to find these overlaps analytically. Here we develop a numerical approach to preferentially generate the states with high overlaps for a quantum quench starting from the ground state or an excited state of an initial Hamiltonian. We use these preferentially generated states, in combination with a "high overlap states truncation scheme" and a modification of the numerical renormalization group, to compute non-equilibrium dynamics following a quench in the Lieb-Liniger model. The method is non-perturbative, works for reasonable numbers of particles, and applies to both continuum and lattice systems. It can also be easily extended to more complicated scenarios, including those with integrability breaking.