Bootstrapping Bloch bands

Abstract

Bootstrap methods, initially developed for solving statistical and quantum field theories, have recently been shown to capture the discrete spectrum of quantum mechanical problems, such as the single particle Schrödinger equation with an anharmonic potential. The core of bootstrap methods builds on exact recursion relations of arbitrary moments of some quantum operator and the use of an adequate set of positivity criteria. We extend this methodology to models with continuous Bloch band spectra, by considering a single quantum particle in a periodic cosine potential. We find that the band structure can be obtained accurately provided the bootstrap uses moments involving both position and momentum variables. We also introduce several new techniques that can apply generally to other bootstrap studies. First, we devise a trick to reduce by one unit the dimensionality of the search space for the variables parametrizing the bootstrap. Second, we employ statistical techniques to reconstruct the distribution probability allowing to compute observables that are analytic functions of the canonical variables. This method is used to extract the Bloch momentum, a quantity that is not readily available from the bootstrap recursion itself.

Bootstrap Method in Harmonic Oscillator

“…We find that the bootstrap problem in harmonic oscillators reduces to the Dirac's ladder operator problem and is exactly solvable. This result suggests that the bootstrap method might be regarded as a generalization of the ladder operator method, and it might explain why the numerical bootstrap method works in various quantum mechanics problems [3,5,6,7,8,9].…”

“…Recently, the bootstrap analysis in zero [1,2] and one-dimensional systems [3] have been proposed, and they are actively studied in various models [4,5,6,7,8,9]. This method works even in matrix models at N = ∞, which is not possible in Monte-Carlo computations [1,2,3,4].…”

The null bootstrap

“…In the textbooks of quantum mechanics, a standard axiom is that a fundamental Hamiltonian is Hermitian. 18 The Hermiticity assumption ensures that the energy spectrum is real and the Hermitian norm is non-negative. Furthermore, the eigenfunctions are guaranteed to be orthogonal and complete.…”

“…One of the impressive results is the most precise determination of the 3d Ising critical exponents [6][7][8][9]. More recently, the positivity principle 1 has also been applied to the studies of matrix models and quantum mechanical systems [10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”

Bootstrapping PT symmetric Hamiltonians

“…In principle since this procedure requires no further information than symmetries and positivity of the norm, traditionally it has been used to search consistent theories as well as its spectrum [2]. However, in the context of quantum mechanics, since there is no restriction on kind of potential that one can write down, the main focus has been to obtain the spectrum of a given quantum mechanical system [3][4][5][6][7][8][9][10][11][12][13].…”