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arXiv:2209.09919 · Physical review. D/Physical review. D.

Bootstrapping the Kronig-Penney model

Matthew J. Blacker, Arpan Bhattacharyya, Aritra Banerjee

Published Dec 14, 2022• 17 citations

Abstract

Recently, bootstrap methods from conformal field theory have been adapted for studying the energy spectrum of various quantum mechanical systems. In this paper, we consider the application of these methods in obtaining the spectrum from the Schr\"odinger equation with periodic potentials, paying particular attention to the Kronig-Penney model of a particle in a one-dimensional lattice. With an appropriate choice of operator basis involving position and momenta, we find that the bootstrap approach efficiently computes the band gaps of the energy spectrum but has trouble effectively constraining the minimum energy. We show how applying more complex constraints involving higher powers of momenta can potentially remedy such a problem. We also propose an approach for analytically constructing the dispersion relation associated with the Bloch momentum of the system.

DOI: 10.1103/physrevd.106.116008

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Periodic structures are ubiquitous in quantum many-body systems and quantum field theories, ranging from lattice models, to compact spaces, to topological phenomena. However, previous bootstrap studies encountered technical challenges even for one-body periodic problems, such as a failure in determining the accurate dispersion relations for Bloch bands. In this work, we develop a new bootstrap procedure to resolve these issues, which does not make use of positivity constraints. We mainly consider a quantum particle in a periodic cosine potential. The same procedure also applies to a particle on a circle, where the role of the Bloch momentum k is played by the boundary condition or the θ angle. We unify the natural set of operators and the translation operator by a new set of operators { e i n x e i a p p s } . To extract the Bloch momentum k , we further introduce a set of differential equations for ⟨ e i n x e i a p p s ⟩ in the translation parameter a . At some fixed a , the boundary conditions can be determined accurately by analytic bootstrap techniques and matching conditions. After solving the differential equations, we impose certain reality conditions to determine the accurate dispersion relations, as well as the k dependence of other physical quantities. We also investigate the case of noninteger s using the Weyl integral in fractional calculus.

Abstract We test the bootstrap approach for determining the spectrum of one dimensional Hamiltonians. In this paper we focus on problems that have a two parameter search space in the bootstrap approach: the double well and a periodic potential associated with the Mathieu equation. For the double well, we compare the bootstrapped energies with contributions from perturbative and non-perturbative results, finding good agreement. For the periodic potentials, we notice that the bootstrap approach gives the band structure of the periodic potential, but it has trouble finding the quasi-momentum of the system. To make further progress on determining the dispersion relation of the bands, new techniques are needed.