Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2211.03819 · Journal of High Energy Physics

Bootstrapping the gap in quantum spin systems

Colin Oscar Nancarrow, Yuan Xin

Published Aug 11, 2023• 20 citations
arXivHTMLPDF

Abstract

Abstract In this work we report on a new bootstrap method for quantum mechanical problems that closely mirrors the setup from conformal field theory (CFT). We use the equations of motion to develop an analogue of the conformal block expansion for matrix elements and impose crossing symmetry in order to place bounds on their values. The method can be applied to any quantum mechanical system with a local Hamiltonian, and we test it on an anharmonic oscillator model as well as the (1 + 1)-dimensional transverse field Ising model (TFIM). For the anharmonic oscillator model we show that a small number of crossing equations provides an accurate solution to the spectrum and matrix elements. For the TFIM we show that the Hamiltonian equations of motion, translational invariance and global symmetry selection rules imposes a rigorous bound on the gap and the matrix elements of TFIM in the thermodynamic limit. The bound improves as we consider larger systems of crossing equations, ruling out more finite-volume solutions. Our method provides a way to probe the low energy spectrum of an infinite lattice from the Hamiltonian rigorously and without approximation.

DOI
10.1007/jhep08(2023)052

Related papers

Bootstrapping Lieb-Schultz-Mattis anomalies

Ryan A. Lanzetta, Lukasz Fidkowski

We incorporate the microscopic assumptions that lead to a certain generalization of the Lieb-Schultz-Mattis (LSM) theorem for one-dimensional spin chains into the conformal bootstrap. Our approach accounts for the ``LSM anomaly'' possessed by these spin chains through a combination of modular bootstrap and correlator bootstrap of symmetry defect operators. Specifically, we obtain universal bounds on the local operator content of $(1+1)d$ conformal field theories (CFTs) that could describe translationally invariant lattice Hamiltonians with a ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N}$ symmetry realized projectively at each site. We assume, for such models, that in the CFT the lattice translation symmetry is realized as an emanant internal ${\mathbb{Z}}_{N}$ symmetry. We present bounds on local operators both with and without refinement by their global symmetry representations. Interestingly, we can obtain nontrivial bounds on charged operators when $N$ is odd, which turns out to be impossible with modular bootstrap alone. Our bounds exhibit distinctive kinks, some of which are approximately saturated by known theories and others that are unexplained. We discuss additional scenarios with the properties necessary for our bounds to apply, including certain multicritical points between $(1+1)d$ symmetry protected topological phases, where we argue that the anomaly studied in our bootstrap calculations should emerge.

arXiv:2207.05092•Physical review. B./Physical review. B•May 19, 2023•13 citations

Bootstrapping Disordered Quantum Systems

Yaprak Önder, Michael G. Scheer, Minjae Cho +1 more

We present a bootstrap framework that yields rigorous two-sided bounds on ground-state observables of quantum systems with quenched disorder. The key idea is to extend the operator algebra to include classical disorder variables and to encode the disorder distribution through a set of moments. The method directly bounds disorder-averaged expectation values for finite or infinite lattice systems and for arbitrary disorder distributions. In contrast to conventional approaches, we do not need to sample disorder realizations, which enables us to make thorough use of symmetries that hold only on average rather than for each disorder realization. Additionally, we show that multi-replica operators can be used to access disorder fluctuations such as the variance of an observable. We apply the method to the one-dimensional random transverse-field Ising model (RTFIM) with discrete, uniform, and Gaussian disorder, obtaining tight bounds on the disorder-averaged ground-state energy density and short-range spin-spin correlators, as well as bounds on the disorder variance of the energy density. Additionally, we show that the difference between the upper and lower bounds on the disorder-averaged ground-state energy density, which we term the bootstrap gap, can be used to map out the phase diagram of the RTFIM. The bootstrap gap is small and flat within the paramagnetic and ferromagnetic phases at weak disorder, grows upon entering the Griffiths regions, and peaks along the critical line.

arXiv:2609.20916•Sep 17, 2026•0 citations

Bootstrapping Conformal Field Theories with the Extremal Functional Method

Sheer El-Showk, Miguel F. Paulos

The existence of a positive linear functional acting on the space of (differences between) conformal blocks has been shown to rule out regions in the parameter space of conformal field theories (CFTs). We argue that at the boundary of the allowed region the extremal functional contains, in principle, enough information to determine the dimensions and operator product expansion (OPE) coefficients of an infinite number of operators appearing in the correlator under analysis. Based on this idea we develop the extremal functional method (EFM), a numerical procedure for deriving the spectrum and OPE coefficients of CFTs lying on the boundary (of solution space). We test the EFM by using it to rederive the low lying spectrum and OPE coefficients of the two-dimensional Ising model based solely on the dimension of a single scalar quasiprimary--no Virasoro algebra required. Our work serves as a benchmark for applications to more interesting, less known CFTs in the near future.

arXiv:1211.2810•Physical Review Letters•Dec 9, 2013•249 citations

Aplicações do Bootstrap Funcional Extremo

Alexander Meinke

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.

•Nov 13, 2018•0 citations

Applications of the Extremal Functional Bootstrap

Alexander Meinke

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.

•Nov 26, 2018•0 citations

Bootstrap bounds for Quantum Spin Systems using String Operators

Nisarg Chadha, Michael G. Scheer, Eslam Khalaf

Bootstrap is a numerical many-body method that provides rigorous bounds on ground-state observables by imposing a set of necessary constraints on the expectation values of operators. The quality of the resulting bounds is sensitive to the choice of operators entering the constraints. In particular, bounds on ground-state correlations are often loose in spontaneous symmetry-breaking (SSB) phases, since local operator sets cannot exclude domain-wall excitations. In this work, we introduce non-local, string-like operators into the bootstrap and show that the program can be formulated directly in thermodynamic limit. We then apply our construction to several 1D spin models. First, we obtain a significant tightening of the bounds in the SSB phase of the 1D transverse-field Ising model. Using the 1D axial next-nearest-neighbor Ising model, we further show that this tightening allows for a quantitative estimate of the locations of phase boundaries. Finally, we generalize the string operators to the 1D $\mathbb{Z}_3$ chiral clock model and track the behavior of the bounds across the phase diagram. Our results broaden the class of constraints available to the bootstrap and open a route toward bootstrapping more general symmetry-broken and topological phases, where the relevant constraints may involve non-local or extended operators.

arXiv:2606.06584•Jun 4, 2026•0 citations
Abstract In this work we report on a new bootstrap method for quantum mechanical problems that closely mirrors the setup from conformal field theory (CFT). We use the equations of motion to develop an analogue of the conformal block expansion for matrix elements and impose crossing symmetry in order to place bounds on their values. The method can be applied to any quantum mechanical system with a local Hamiltonian, and we test it on an anharmonic oscillator model as well as the (1 + 1)-dimensional transverse field Ising model (TFIM). For the anharmonic oscillator model we show that a small number of crossing equations provides an accurate solution to the spectrum and matrix elements. For the TFIM we show that the Hamiltonian equations of motion, translational invariance and global symmetry selection rules imposes a rigorous bound on the gap and the matrix elements of TFIM in the thermodynamic limit. The bound improves as we consider larger systems of crossing equations, ruling out more finite-volume solutions. Our method provides a way to probe the low energy spectrum of an infinite lattice from the Hamiltonian rigorously and without approximation.
We incorporate the microscopic assumptions that lead to a certain generalization of the Lieb-Schultz-Mattis (LSM) theorem for one-dimensional spin chains into the conformal bootstrap. Our approach accounts for the ``LSM anomaly'' possessed by these spin chains through a combination of modular bootstrap and correlator bootstrap of symmetry defect operators. Specifically, we obtain universal bounds on the local operator content of (1+1)d(1+1)d(1+1)d conformal field theories (CFTs) that could describe translationally invariant lattice Hamiltonians with a ${\mathbb{Z}}_{N}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{N}$ symmetry realized projectively at each site. We assume, for such models, that in the CFT the lattice translation symmetry is realized as an emanant internal ZN{\mathbb{Z}}_{N}ZN​ symmetry. We present bounds on local operators both with and without refinement by their global symmetry representations. Interestingly, we can obtain nontrivial bounds on charged operators when NNN is odd, which turns out to be impossible with modular bootstrap alone. Our bounds exhibit distinctive kinks, some of which are approximately saturated by known theories and others that are unexplained. We discuss additional scenarios with the properties necessary for our bounds to apply, including certain multicritical points between (1+1)d(1+1)d(1+1)d symmetry protected topological phases, where we argue that the anomaly studied in our bootstrap calculations should emerge.
We present a bootstrap framework that yields rigorous two-sided bounds on ground-state observables of quantum systems with quenched disorder. The key idea is to extend the operator algebra to include classical disorder variables and to encode the disorder distribution through a set of moments. The method directly bounds disorder-averaged expectation values for finite or infinite lattice systems and for arbitrary disorder distributions. In contrast to conventional approaches, we do not need to sample disorder realizations, which enables us to make thorough use of symmetries that hold only on average rather than for each disorder realization. Additionally, we show that multi-replica operators can be used to access disorder fluctuations such as the variance of an observable. We apply the method to the one-dimensional random transverse-field Ising model (RTFIM) with discrete, uniform, and Gaussian disorder, obtaining tight bounds on the disorder-averaged ground-state energy density and short-range spin-spin correlators, as well as bounds on the disorder variance of the energy density. Additionally, we show that the difference between the upper and lower bounds on the disorder-averaged ground-state energy density, which we term the bootstrap gap, can be used to map out the phase diagram of the RTFIM. The bootstrap gap is small and flat within the paramagnetic and ferromagnetic phases at weak disorder, grows upon entering the Griffiths regions, and peaks along the critical line.
The existence of a positive linear functional acting on the space of (differences between) conformal blocks has been shown to rule out regions in the parameter space of conformal field theories (CFTs). We argue that at the boundary of the allowed region the extremal functional contains, in principle, enough information to determine the dimensions and operator product expansion (OPE) coefficients of an infinite number of operators appearing in the correlator under analysis. Based on this idea we develop the extremal functional method (EFM), a numerical procedure for deriving the spectrum and OPE coefficients of CFTs lying on the boundary (of solution space). We test the EFM by using it to rederive the low lying spectrum and OPE coefficients of the two-dimensional Ising model based solely on the dimension of a single scalar quasiprimary--no Virasoro algebra required. Our work serves as a benchmark for applications to more interesting, less known CFTs in the near future.
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.
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.
Bootstrap is a numerical many-body method that provides rigorous bounds on ground-state observables by imposing a set of necessary constraints on the expectation values of operators. The quality of the resulting bounds is sensitive to the choice of operators entering the constraints. In particular, bounds on ground-state correlations are often loose in spontaneous symmetry-breaking (SSB) phases, since local operator sets cannot exclude domain-wall excitations. In this work, we introduce non-local, string-like operators into the bootstrap and show that the program can be formulated directly in thermodynamic limit. We then apply our construction to several 1D spin models. First, we obtain a significant tightening of the bounds in the SSB phase of the 1D transverse-field Ising model. Using the 1D axial next-nearest-neighbor Ising model, we further show that this tightening allows for a quantitative estimate of the locations of phase boundaries. Finally, we generalize the string operators to the 1D Z3\mathbb{Z}_3Z3​ chiral clock model and track the behavior of the bounds across the phase diagram. Our results broaden the class of constraints available to the bootstrap and open a route toward bootstrapping more general symmetry-broken and topological phases, where the relevant constraints may involve non-local or extended operators.