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arXiv:2106.14027 · Physical review. B./Physical review. B

Efficient impurity-bath trial states from superposed Slater determinants

Izak Snyman, Serge Florens

Published Nov 19, 2021• 5 citations
arXivHTMLPDF

Abstract

The representation of ground states of fermionic quantum impurity problems as superpositions of Gaussian states has recently been given a rigorous mathematical foundation [S. Bravyi and D. Gosset, Commun. Math. Phys. 356, 451 (2017)]. It is natural to ask how many parameters are required for an efficient variational scheme based on this representation. An upper bound is $O({N}^{2})$, where $N$ is the system size, which corresponds to the number parameters needed to specify an arbitrary Gaussian state. We provide an alternative representation, with more favorable scaling, only requiring $O(N)$ parameters, that we illustrate for the interacting resonant-level model. We achieve the reduction by associating mean-field-like parent Hamiltonians with the individual terms in the superposition, using physical insight to retain only the most relevant channels in each parent Hamiltonian. We benchmark our variational ansatz against the numerical renormalization group, and compare our results to other variational schemes of a similar nature to ours. Apart from the ground-state energy, we also study the spectrum of the covariance matrix---a very stringent measure of accuracy. Our approach outperforms some existing variational schemes and remains quantitatively accurate in the numerically challenging near-critical regime.

DOI
10.1103/physrevb.104.195136

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The representation of ground states of fermionic quantum impurity problems as superpositions of Gaussian states has recently been given a rigorous mathematical foundation [S. Bravyi and D. Gosset, Commun. Math. Phys. 356, 451 (2017)]. It is natural to ask how many parameters are required for an efficient variational scheme based on this representation. An upper bound is O(N2)O({N}^{2})O(N2), where NNN is the system size, which corresponds to the number parameters needed to specify an arbitrary Gaussian state. We provide an alternative representation, with more favorable scaling, only requiring O(N)O(N)O(N) parameters, that we illustrate for the interacting resonant-level model. We achieve the reduction by associating mean-field-like parent Hamiltonians with the individual terms in the superposition, using physical insight to retain only the most relevant channels in each parent Hamiltonian. We benchmark our variational ansatz against the numerical renormalization group, and compare our results to other variational schemes of a similar nature to ours. Apart from the ground-state energy, we also study the spectrum of the covariance matrix---a very stringent measure of accuracy. Our approach outperforms some existing variational schemes and remains quantitatively accurate in the numerically challenging near-critical regime.
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 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.
Predicting the properties of strongly correlated materials is a significant challenge in condensed matter theory. The widely used dynamical mean-field theory faces difficulty in solving quantum impurity models numerically. Hybrid quantum-classical algorithms such as variational quantum eigensolvers emerge as a potential solution for quantum impurity models. A common challenge in these algorithms is the rapid growth of the number of variational parameters with the number of spin-orbitals in the impurity. In our approach to this problem, we develop compact using a combination of two different strategies. First, we employ a compact physics-inspired , k -unitary cluster Jastrow , developed in the field of quantum chemistry. Second, we eliminate largely redundant variational parameters of physics-inspired associated with bath sites based on physical intuition. This is based on the fact that a quantum impurity model with a star-like geometry has no direct hopping between bath sites. We benchmark the accuracy of these for both ground-state energy and dynamic quantities by solving typical quantum impurity models with and without shot noise. The results suggest that we can maintain the accuracy of ground-state energy while we drop the number of variational parameters associated with bath sites. Furthermore, we demonstrate that a moment expansion, when combined with the proposed , can calculate the imaginary-time Green's functions under the influence of shot noise. This study demonstrates the potential for addressing complex impurity models in large-scale quantum simulations with fewer variational parameters without sacrificing accuracy. Published by the American Physical Society 2024
We develop a variational scheme called the ``Gutzwiller renormalization group'' (GRG), which enables us to calculate the ground state of Anderson impurity models (AIM) with arbitrary numerical precision. Our method exploits the low-entanglement property of the ground state of local Hamiltonians in combination with the framework of the Gutzwiller wave function and indicates that the ground state of the AIM has a very simple structure, which can be represented very accurately in terms of a surprisingly small number of variational parameters. We perform benchmark calculations of the single-band AIM that validate our theory and suggest that the GRG might enable us to study complex systems beyond the reach of the other methods presently available and pave the way to interesting generalizations, e.g., to nonequilibrium transport in nanostructures.
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.
A neural-network approach to optimize the selection of Slater determinants in configuration interaction calculations for condensed-matter quantum many-body systems is developed. We exemplify our algorithm on the discrete version of the single-impurity Anderson model with up to 299 bath sites. Employing a neural network classifier and active learning, our algorithm enhances computational efficiency by iteratively identifying the most relevant Slater determinants for the ground state wave function. We benchmark our results against established methods and investigate the efficiency of our approach compared to another basis truncation scheme without a neural network. Our algorithm demonstrates a substantial improvement in the efficiency of determinant selection, yielding a more compact and computationally manageable basis without compromising accuracy. Given the straightforward application of our neural-network-supported selection scheme to other model Hamiltonians of quantum many-body clusters, our algorithm can significantly advance selective configuration interaction calculations in the context of correlated condensed matter.