Beyond the Gap: What Imaginary-Time Evolution Reveals About the Edge of the Continuum
Abstract
The spectral gap, the energy separating the ground state from the first excitation, governs many quantum phenomena, yet it is far harder to compute than the ground state. We propose a mathematically rigorous method for extracting the gap from imaginary-time evolution in infinite-dimensional Hamiltonians, where the first excitation typically marks the edge of a continuous spectrum. We show that a gap is not automatically detected: detection depends on how the initial state and observable couple the ground state to excitations at the bottom of the continuum. When this coupling follows a power law, the gap is recovered together with a logarithmic correction whose coefficient yields the exponent of the spectral density near the edge of the continuum, which reflects the physics of the excitations. We confirm both predictions numerically for an exactly solvable model. Our results offer a new numerical approach to many-body quantum physics, in which both the gap and the threshold exponent are obtained merely at the cost of a ground-state calculation.
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BibTeX
@misc{karapetyan2026beyond,
title = {{Beyond the Gap: What Imaginary-Time Evolution Reveals About the Edge of the Continuum}},
author = {Ruben Karapetyan and Denys Bondar and Jakub Mareček and Aleš Wodecki},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06454},
eprint = {2611.06454},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06454},
}
RIS
TY - UNPB
TI - Beyond the Gap: What Imaginary-Time Evolution Reveals About the Edge of the Continuum
AU - Ruben Karapetyan
AU - Denys Bondar
AU - Jakub Mareček
AU - Aleš Wodecki
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06454
UR - https://arxiv.org/abs/2611.06454
AB - The spectral gap, the energy separating the ground state from the first excitation, governs many quantum phenomena, yet it is far harder to compute than the ground state. We propose a mathematically rigorous method for extracting the gap from imaginary-time evolution in infinite-dimensional Hamiltonians, where the first excitation typically marks the edge of a continuous spectrum. We show that a gap is not automatically detected: detection depends on how the initial state and observable couple the ground state to excitations at the bottom of the continuum. When this coupling follows a power law, the gap is recovered together with a logarithmic correction whose coefficient yields the exponent of the spectral density near the edge of the continuum, which reflects the physics of the excitations. We confirm both predictions numerically for an exactly solvable model. Our results offer a new numerical approach to many-body quantum physics, in which both the gap and the threshold exponent are obtained merely at the cost of a ground-state calculation.
ER -