Bootstrapping the Kronig-Penney model

Blacker, Matthew J.; Bhattacharyya, Arpan; Banerjee, Aritra

doi:10.1103/physrevd.106.116008

Phys. Rev. D

2022

DOI: 10.1103/physrevd.106.116008

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Bootstrapping the Kronig-Penney model

Matthew J. Blacker

Arpan Bhattacharyya²,

Aritra Banerjee³

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Cited by 6 publications

References 21 publications

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Bootstrapping the gap in quantum spin systems

Nancarrow,

Xin

2023

J. High Energ. Phys.

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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.

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Bootstrapping the gap in quantum spin systems

Nancarrow,

Xin

2023

J. High Energ. Phys.

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Bootstrapping the Abelian lattice gauge theories

Li,

Zhou

2024

J. High Energ. Phys.

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We study the ℤ2 and U(1) Abelian lattice gauge theories using a bootstrap method, in which the loop equations and positivity conditions are employed for Wilson loops with lengths L ⩽ Lmax to derive two-sided bounds on the Wilson loop averages. We address a fundamental question that whether the constraints from loop equations and positivity are strong enough to solve lattice gauge theories. We answer this question by bootstrapping the 2D U(1) lattice gauge theory. We show that with sufficiently large Lmax = 60, the two-sided bounds provide estimates for the plaquette averages with precision near 10−8 or even higher, suggesting the bootstrap constraints are sufficient to numerically pin down this theory. We compute the bootstrap bounds on the plaquette averages in the 3D ℤ2 and U(1) lattice gauge theories with Lmax = 16. In the regions with weak or strong coupling, the two-sided bootstrap bounds converge quickly and coincide with the perturbative results to high precision. The bootstrap bounds are well consistent with the Monte Carlo results in the nonperturbative region. We observe interesting connections between the bounds generated by the bootstrap computations and the Griffiths’ inequalities. We present results towards bootstrapping the string tension and glueball mass in Abelian lattice gauge theories.

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Unify the Effect of Anharmonicity in Double-Wells and Anharmonic Oscillators

Fan,

Zhang,

2024

Int J Theor Phys

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Bootstrapping the Kronig-Penney model

Cited by 6 publications

References 21 publications

Bootstrapping the gap in quantum spin systems

Bootstrapping the gap in quantum spin systems

Bootstrapping the Abelian lattice gauge theories

Unify the Effect of Anharmonicity in Double-Wells and Anharmonic Oscillators

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