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COLD ATOMS AND THE CONFORMAL BOOTSTRAP

Joshua William Squires

Published Jan 1, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.7298/vvt1-jb31

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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.
In this dissertation we describe progress towards carving out the space of non-integrable conformal field theories (CFTs) in 1+1 dimensions using conformal bootstrap techniques. We develop and apply numerical and analytical techniques to explore the non-perturbative consequences of conformal symmetry, unitarity and locality on the spectrum and dynamics of 2D CFTs. Throughout we focus on compact irrational CFTs, which have a discrete spectrum consisting of an infinite number of primary operators and exhibit chaotic dynamics. In the first part of the dissertation we report on numerical advances in charting the space of 2D CFTs. We use semidefinite programming to compute bounds on the gap in the spectrum of primary operators as a function of the central charge and on spectral functions that capture the spectral density of primary operators. We then develop recursion relations for arbitrary Virasoro conformal blocks, and explore the constraints of modular invariance of the genus-two partition function, leading to ``critical surfaces'' that bound the heavy structure constants of a CFT. In the second part of the dissertation we develop the analytic bootstrap program for two dimensional conformal field theories, focusing on universal aspects of irrational unitary CFTs and holography. We show that the large-spin sector of irrational CFTs is universally governed by a ``Virasoro mean field theory,'' incorporating exact stress tensor dynamics into the analytic bootstrap and leading to non-perturbative insights about the bound-state dynamics of three-dimensional quantum gravity. We then leverage a generalized notion of modular invariance to derive a universal asymptotic formula for the average value of the structure constants whenever one or more of the operators has large conformal dimension. The large central charge limit makes contact with quantum gravity in three dimensions, where the averaging over heavy states corresponds to coarse-graining over black hole microstates. Finally, we study the Euclidean gravitational path integral of pure AdS3 quantum gravity, and use a modular bootstrap analysis to suggest new saddle point configurations that can be consistently included in the gravitational path integral and whose inclusion renders the dual torus partition function unitary.
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.
Breaking Symmetries: N=2\mathcal{N} = 2N=2 Spin Chains and Conformal Field Theory at Finite Temperature
Conformal field theories (CFTs) and supersymmetric field theories constitute special classes of quantum field theories with applications ranging from critical phenomena and confinement to quantum gravity via holography. Their large symmetry groups make them ideal testing grounds for theoretical developments, enabling the use of powerful and elegant methods. Among these, conformal bootstrap and integrability have proven particularly successful. In this thesis, we take steps toward extending these methods beyond their conventional domains by breaking conformal symmetry through finite temperature effects and reducing supersymmetry via orbifolding. For spin chains arising from N=2\mathcal{N} = 2N=2 superconformal field theories obtained by orbifolding N=4\mathcal{N} = 4N=4 super Yang–Mills theory in four dimensions, we generalize the coordinate Bethe ansatz by incorporating long-range corrections.On the conformal bootstrap side, after breaking conformality by placing the CFT on a d-dimensional cylinder, we develop an analytic framework based on dispersion relations. We apply this method to several theories, including large-N CFTs in four dimensions. Furthermore, we combine this analytic approach with existing numerical bootstrap techniques at finite temperature to compute new thermal correlation functions in the three-dimensional Ising CFT.
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.
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.
A novel formalism of quantum statistical mechanics, based on the zero-temperature S-matrix of the quantum system, is presented in this thesis. In our new formalism, the lowest order approximation ("two-body approximation") corresponds to the exact resummation of all binary collision terms, and can be expressed as an integral equation reminiscent of the thermodynamic Bethe Ansatz (TBA). Two applications of this formalism are explored: the critical point of a weakly-interacting Bose gas in two dimensions, and the scaling behavior of quantum gases at the unitary limit in two and three spatial dimensions. We found that a weakly-interacting 2D Bose gas undergoes a superfluid transition at Tc [ALMOST EQUAL TO] 2[pi]n/[m log(2[pi]/mg )], where n is the number density, m the mass of a particle, and g the coupling. In the unitary limit where the coupling g diverges, the two-body kernel of our integral equation has simple forms in both two and three spatial dimensions, and we were able to solve the integral equation numerically. Various scaling functions in the unitary limit are defined (as functions of [MICRO SIGN]/T ) and computed from the numerical solutions. For bosons in three spatial dimensions, we present evidence that the gas undergoes 3 a strongly interacting version of Bose-Einstein condensation at n[lamda]T [ALMOST EQUAL TO] 1.3, where n is the number density and [lamda]T is the thermal wavelength. Finally, we look at the ratio of shear viscosity to entropy density [eta]/s of the unitary quantum gas, which has a conjectured lower bound of [eta]/s [GREATER-THAN OR EQUAL TO] /4[pi]kb based on the AdS/CFT correspondence of a strongly coupled Yang-Mills theory.