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arXiv:2109.06600 · Journal of Physics A Mathematical and Theoretical

Bootstrapping Bloch bands

S. Tchoumakov, Serge Florens

Published Nov 23, 2021• 27 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/ac3c82

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•Nov 26, 2018•0 citations
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.
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.
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.
We solve the Schrödinger equation for a particle in the circle with a quantum mechanical bootstrap method. The main idea is that in physical situations one needs solutions of the recursion relations for the Fourier coefficients of the measure that vanish exponentially fast. We show that this results in a (generalized) continued fraction representation of the solution. The solution is an analytical function of the couplings and the energy. We also show how one this method directly computes the density of states and checking that the answer is physical is reduced to checking an inequality for a single variable. We also show how to reconstruct the wave function and how to extract the quasimomentum of the particle as a function of the energy. We illustrate our solution in the simplest case of the Mathieu equation, where we show that the method can easily resolve exponentially small bands in the band spectrum.
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.
General positivity constraints linking various powers of observables in energy eigenstates can be used to sharply locate acceptable regions for the energy eigenvalues, provided that efficient recursive methods are available to calculate the matrix elements. These recursive methods are derived by looking at the commutation relations of the observables with the Hamiltonian. We discuss how this self-consistent (bootstrap) approach can be applied to the study of digitized scalar field theory in the harmonic basis. Using known results, we develop the method by testing on quantum systems, including the harmonic and anharmonic oscillators. We report recent numerical results for up to four coupled anharmonic oscillators. From here, we consider the possibility of using the groundwork of this method as a means of studying phase transitions in 1+1 dimensions.
The study of conformal symmetry is motivated through an example in statistical mechanics and then rigorously developed in quantum field theories in general spatial dimensions.In particular, primary fields are introduced as the fundamental objects of such theories and then studied in the formalism of radial quantization.The implications of conformal invariance on the functional form of correlation functions are studied in detail.Conformal blocks are defined and various approaches to their analytical and numerical calculation are presented with a special emphasis on the one-dimensional case.Building on these preliminaries, a modern formulation of the conformal bootstrap program and its various extensions are discussed.Examples are given in which bounds on the scaling dimensions in a one-dimensional theory are derived numerically.Using these results I motivate the technique of using the extremal functional bootstrap which I then develop in more detail.Many technical details are discussed and examples shown.After a brief discussion of conformal field theories with a boundary I apply numerical methods to find constraints on the spectrum of the 3D Ising model.Another application is presented in which I study the 4-point function on the boundary of a particular theory in Anti-de-Sitter space in order to approximate the mass spectrum of the theory.