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arXiv:1802.03861 · Physical review. B./Physical review. B

Variational principle for quantum impurity systems in and out of equilibrium: Application to Kondo problems

Yuto Ashida, Tao Shi, Mari Carmen Bañuls, J. I. Cirac, Eugene Demler

Published Jul 11, 2018• 37 citations
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Abstract

We provide a detailed formulation of the recently proposed variational approach [Y. Ashida, T. Shi, M.-C. Ba\~nuls, J. I. Cirac, and E. Demler, Phys. Rev. Lett. 121, 026805 (2018)] to study ground-state properties and out-of-equilibrium dynamics for generic quantum spin-impurity systems. Motivated by the original ideas of Tomonaga, Lee, Low, and Pines, we construct a canonical transformation that completely decouples the impurity from the bath degrees of freedom. By combining this transformation with a Gaussian ansatz for the fermionic bath, we obtain a family of variational many-body states that can efficiently encode the strong entanglement between the impurity and fermions of the bath. We give a detailed derivation of equations of motions in the imaginary- and real-time evolutions on the variational manifold. We benchmark our approach by applying it to investigate ground-state and dynamical properties of the anisotropic Kondo model and compare results with those obtained using the matrix-product state (MPS) ansatz. We show that our approach can achieve an accuracy comparable to MPS-based methods with several orders of magnitude fewer variational parameters than the corresponding MPS ansatz. Comparisons to the Yosida ansatz and the exact solution from the Bethe ansatz are also discussed. We use our approach to investigate the two-lead Kondo model and analyze its long-time spatiotemporal behavior and the conductance behavior at finite bias and magnetic fields. The obtained results are consistent with the previous findings in the Anderson model and the exact solutions at the Toulouse point.

DOI
10.1103/physrevb.98.024103

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We provide a detailed formulation of the recently proposed variational approach [Y. Ashida, T. Shi, M.-C. Ba~nuls, J. I. Cirac, and E. Demler, Phys. Rev. Lett. 121, 026805 (2018)] to study ground-state properties and out-of-equilibrium dynamics for generic quantum spin-impurity systems. Motivated by the original ideas of Tomonaga, Lee, Low, and Pines, we construct a canonical transformation that completely decouples the impurity from the bath degrees of freedom. By combining this transformation with a Gaussian ansatz for the fermionic bath, we obtain a family of variational many-body states that can efficiently encode the strong entanglement between the impurity and fermions of the bath. We give a detailed derivation of equations of motions in the imaginary- and real-time evolutions on the variational manifold. We benchmark our approach by applying it to investigate ground-state and dynamical properties of the anisotropic Kondo model and compare results with those obtained using the matrix-product state (MPS) ansatz. We show that our approach can achieve an accuracy comparable to MPS-based methods with several orders of magnitude fewer variational parameters than the corresponding MPS ansatz. Comparisons to the Yosida ansatz and the exact solution from the Bethe ansatz are also discussed. We use our approach to investigate the two-lead Kondo model and analyze its long-time spatiotemporal behavior and the conductance behavior at finite bias and magnetic fields. The obtained results are consistent with the previous findings in the Anderson model and the exact solutions at the Toulouse point.
A versatile and efficient variational approach is developed to solve in- and out-of-equilibrium problems of generic quantum spin-impurity systems. Employing the discrete symmetry hidden in spin-impurity models, we present a new canonical transformation that completely decouples the impurity and bath degrees of freedom. Combining it with Gaussian states, we present a family of many-body states to efficiently encode nontrivial impurity-bath correlations. We demonstrate its successful application to the anisotropic and two-lead Kondo models by studying their spatiotemporal dynamics and universal behavior in the correlations, relaxation times, and the differential conductance. We compare them to previous analytical and numerical results. In particular, we apply our method to study new types of nonequilibrium phenomena that have not been studied by other methods, such as long-time crossover in the ferromagnetic easy-plane Kondo model. The present approach will be applicable to a variety of unsolved problems in solid-state and ultracold-atomic systems.
The coherent superposition of nonorthogonal fermionic Gaussian states has been shown to be an efficient approximation to the ground states of quantum impurity problems [Bravyi and Gosset, Commun. Math. Phys. 356, 451 (2017)]. We propose a practical approach for performing a variational calculation based on such states. Our method is based on approximate imaginary-time equations of motion that decouple the dynamics of each Gaussian state forming the ansatz. It is independent of the lattice connectivity of the model and the implementation is highly parallelizable. To benchmark our variational method, we calculate the spin-spin correlation function and R\'enyi entanglement entropy of an Anderson impurity, allowing us to identify the screening cloud and compare to density matrix renormalization group calculations. Secondly, we study the screening cloud of the two-channel Kondo model, a problem difficult to tackle using existing numerical tools.
We present a new variational method for investigating the ground state and out of equilibrium dynamics of quantum many-body bosonic and fermionic systems. Our approach is based on constructing variational wavefunctions which extend Gaussian states by including generalized canonical transformations between the fields. The key advantage of such states compared to simple Gaussian states is presence of non-factorizable correlations and the possibility of describing states with strong entanglement between particles. In contrast to the commonly used canonical transformations, such as the polaron or Lang-Firsov transformations, we allow parameters of the transformations to be time dependent, which extends their regions of applicability. We derive equations of motion for the parameters characterizing the states both in real and imaginary time using the differential structure of the variational manifold. The ground state can be found by following the imaginary time evolution until it converges to a steady state. Collective excitations in the system can be obtained by linearizing the real-time equations of motion in the vicinity of the imaginary time steady-state solution. Our formalism allows us not only to determine the energy spectrum of quasiparticles and their lifetime, but to obtain the complete spectral functions and to explore far out of equilibrium dynamics such as coherent evolution following a quantum quench. We illustrate and benchmark this framework with several examples: a single polaron in the Holstein and Su-Schrieer-Heeger models, non-equilibrium dynamics in the spin-boson and Kondo models, the superconducting to charge density wave phase transitions in the Holstein model.
The Kondo model describes a magnetic impurity in a metallic host. Here, the authors use variational approaches (Lanczos, Yosida, Gutzwiller) and quasi-exact numerical techniques (numerical renormalization group, density matrix renormalization group) to study the ground-state energy and magnetization, and compare these properties to Bethe ansatz results. The Gutzwiller wave function is an excellent trial state for strong coupling. For weak coupling, the numerical renormalization group gives the best description of the magnetic properties. A combination of Bethe ansatz and perturbation theory provides new benchmark results for the zero-field susceptibility.
We present a numerical method for the study of correlated quantum impurity problems out of equilibrium, which is particularly suited to address steady-state properties within dynamical mean field theory. The approach, recently introduced by Arrigoni et al. [Phys. Rev. Lett. 110, 086403 (2013)], is based upon a mapping of the original impurity problem onto an auxiliary open quantum system, consisting of the interacting impurity coupled to bath sites as well as to a Markovian environment. The dynamics of the auxiliary system is governed by a Lindblad master equation whose parameters are used to optimize the mapping. The accuracy of the results can be readily estimated and systematically improved by increasing the number of auxiliary bath sites, or by introducing a linear correction. Here, we focus on a detailed discussion of the proposed approach including technical remarks. To solve for the Green's functions of the auxiliary impurity problem, a non-Hermitian Lanczos diagonalization is applied. As a benchmark, results for the steady-state current-voltage characteristics of the single-impurity Anderson model are presented. Furthermore, the bias dependence of the single-particle spectral function and the splitting of the Kondo resonance are discussed. In its present form, the method is fast, efficient, and features a controlled accuracy.