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International Conference on Pervasive Computational Technologies (ICPCT)

Quantum Bootstrap in Microcanonical Ensembles: Computational Insights and Applications

Shujaatali Badami

Published Feb 8, 2025• 3 citations
Publisher

Abstract

This paper explores the quantum bootstrap method, a promising numerical approach rooted in self-consistency conditions, and its application in microcanonical ensembles. Initially developed in the 1960s and ’70s, the bootstrap technique has found applications in various fields, including quantum mechanics, lattice theory, conformal field theory, and matrix models. While extensively studied in quantum mechanics, this paper introduces a classical correspondence to the quantum bootstrap method, expanding its applicability to microcanonical ensembles. The study incorporates a novel approach, demonstrating stronger constraints by integrating additional information in phase space. Numerical examples, including the double well potential, harmonic oscillator, Coulomb potential, and non-relativistic Toda model, illustrate the method’s effectiveness in resolving unphysical energy regions and improving computational precision. These results highlight its potential as a powerful tool in computational physics and quantum computing.

DOI
10.1109/icpct64145.2025.10939270

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This paper explores the quantum bootstrap method, a promising numerical approach rooted in self-consistency conditions, and its application in microcanonical ensembles. Initially developed in the 1960s and ’70s, the bootstrap technique has found applications in various fields, including quantum mechanics, lattice theory, conformal field theory, and matrix models. While extensively studied in quantum mechanics, this paper introduces a classical correspondence to the quantum bootstrap method, expanding its applicability to microcanonical ensembles. The study incorporates a novel approach, demonstrating stronger constraints by integrating additional information in phase space. Numerical examples, including the double well potential, harmonic oscillator, Coulomb potential, and non-relativistic Toda model, illustrate the method’s effectiveness in resolving unphysical energy regions and improving computational precision. These results highlight its potential as a powerful tool in computational physics and quantum computing.
We study the quantum-mechanical bootstrap as it applies to the bound states of several central potentials in three dimensions. As part of this effort, we show how the bootstrap approach may be applied to ``non-algebraic'' potentials, such as the Yukawa potential (which asymptotically decays as an exponential) and a Gaussian potential. We additionally review the bootstrap of the Coulomb potential, demonstrate a high-precision bootstrap of the Cornell potential, and study conformal quantum mechanics. These results further recommend the bootstrap as a numerical method for high-precision calculations of ground-state physics, where applicable: for example, we are able to determine the critical coupling in the Cornell potential to better than one part in 10710^7107, the most precise determination to date. Lower bounds on energies are also of high precision, occasionally one part in greater than 10810^8108. Finally, we discuss the circumstances under which we are able to obtain meaningful upper bounds on ground-state energies.
A novel bootstrap method is exemplified by lattice theories and quantum mechanics. Positivity of expectation values combined with sets of non-trivial constraint equations and semidefinite optimization rigorously and precisely bound correlation functions. Starting with the example of a one-dimensional integral, the technique is built upon to extract limit cycles in a two-dimensional classical dynamical system and spin correlation functions in the Statistical Ising Model. Further positivity constraints and simplifications arising in a large rank limit are implemented to bound the energy expectation value as a function of temperature in Matrix Quantum Mechanics. Along the way, the S-matrix of Chern-Simons-matter theory is computed as an example of computational simplification at large gauge group rank.
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.
Path integral representations for the quantum microcanonical ensemble are presented. In the quantum microcanonical ensemble, two operators are of primary interest. First, $\stackrel{^}{\ensuremath{\rho}}=\ensuremath{\delta}(E\ensuremath{-}\ifmmode \hat{H}\else \^{H}\fi{})$ corresponds to the microcanonical density matrix and can be used to calculate expectation values. Second, $\mathrm{N\ifmmode \hat{}\else \^{}\fi{}}=\ensuremath{\Theta}(E\ensuremath{-}\ifmmode \hat{H}\else \^{H}\fi{})$ can give the number of states with energy En<E.{E}_{n}<E.En​<E. We consider position matrix elements of both of these operators $\ensuremath{\Omega}{(x,x}^{\ensuremath{'}},E)=〈{x}^{\ensuremath{'}}|\ensuremath{\delta}(E\ensuremath{-}\ifmmode \hat{H}\else \^{H}\fi{})|x〉$ and $\ensuremath{\Theta}{(x,x}^{\ensuremath{'}},E)=〈{x}^{\ensuremath{'}}|\ensuremath{\theta}(E\ensuremath{-}\ifmmode \hat{H}\else \^{H}\fi{})|x〉.$ A path integral formalism leads to exact integral representations for $\ensuremath{\Omega}{(x,x}^{\ensuremath{'}},E)$ and $\ensuremath{\Theta}{(x,x}^{\ensuremath{'}},E).$ We present both phase space and configuration space forms. For simple systems, such as the free particle and harmonic oscillator, exact solutions are possible. For more complicated systems, expansion schemes or numerical evaluations are required. A perturbative calculation and numerical integration results are presented for the quantum anharmonic oscillator.
Statistical mechanics naturally lends itself to computational algorithms that use random sampling to leverage the law of large numbers. Though these methods have proven invaluable, in many cases yielding solutions to otherwise intractable problems, to an extent they also obscure the underlying physics. The two numerical methods studied in this thesis do not fit this description. The first, the conformal bootstrap, imposes symmetry constraints on the four-point correlation functions of a conformal field theory to restrict the spectrum of allowed scaling dimensions of a theory. A variation of the conformal bootstrap is implemented to treat two important non-unitary CFTs, percolation and the self-avoiding walk. The second involves a formalism which provides an integral equation with a kernel based on the two-body S-matrix. The solution of this integral equation represents a single particle's energy in the presence of interactions with the rest of a cold atom gas. The formalism is applied to Berezinskii-Kosterlitz-Thouless and upper branch phase transitions.
This thesis explores the application of numerical bootstrap techniques to two-point functions of bulk insertions in the presence of a boundary in two-dimensional conformal field theories. The first part of this thesis is devoted to presenting a comprehensive introduction to essential concepts in the numerical bootstrap of two-dimensional CFTs. Namely, an extensive introduction to the basics of conformal field theories (CFTs), their properties arising from the algebraic structure of conformal transformations, consitency conditions on the observables in a CFT, as well as the effects of the presence of defects in two-dimensional CFTs, is given. This is followed by a quick recount of the numerical techniques employed in the literature to solve for the aforementioned consistency conditions. These chapters form an overview of the seminal works that set the backdrop of the contributions made in the second part of the thesis. In the contributions of the thesis, we discuss the progress made in the investigations in (Meineri and Radhakrishnan 2025) where two-point functions in BCFTs are bootstrapped by treating them as a part of a larger system of consistency conditions involving the partition function on an annulus, the two-point function in the presence of a boundary, and four-point functions on the infinite plane. This perspective of treating the two-point function as a part of a larger system makes accessible parts of the BCFT data that have so-far remained inaccessible, and we present results on the boundary gap in the two-point function, and improvements in existing numerical bounds on the boundary entropy.