Bootstrap certification of string order in quantum spin chains
Abstract
The many-body bootstrap certifies ground-state properties by minimizing energy over correlators constrained only by positivity, without any variational wavefunction. Its cost, however, grows exponentially with the size of the operators involved, placing the long-range string correlators that diagnose topological phases out of reach. We overcome this by treating string operators as primary objects: bare and endpoint-dressed strings satisfy a closed operator algebra with one another and with local words, yielding a semidefinite program whose cost grows only polynomially with string length. A single computation then yields estimates of string correlators of all lengths. Rigorous two-sided bounds are obtained for each target string once the ground-state energy is pinned within a window. When benchmarked against density-matrix renormalization group computations on the cluster Ising and spin−1 Heisenberg chains, the method certifies the nonlocal string order of the cluster and Haldane phases directly from the Hamiltonian. Our rigorous certification of string order parameters reveal that the sharpness of the bounds is set by which operators enter the calculation, not by their length alone. By exploiting algebraic closure of strings and local words, our framework lifts spatially extended observables into polynomially tractable bootstrap variables, opening up a route to wavefunction-free certification of nonlocal order in quantum many-body systems.
Related
- String Order in Quench Dynamics as a Probe of Quantum Phase Transitions
- Analytic Dual Resonance and the String Bootstrap
- IR extremality and UV softness in string bootstrap
- Optimal Entanglement Routing in Quantum Repeater Chains: Beyond Fixed Operation Order and Purification Schedule
- Super Liouville theory on the torus: bootstrap and the super Virasoro minimal string
BibTeX
@misc{banerjee2026bootstrap,
title = {{Bootstrap certification of string order in quantum spin chains}},
author = {Sagnik Banerjee and Haoyu Guo and Debanjan Chowdhury},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22420},
eprint = {2609.22420},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22420},
}
RIS
TY - UNPB
TI - Bootstrap certification of string order in quantum spin chains
AU - Sagnik Banerjee
AU - Haoyu Guo
AU - Debanjan Chowdhury
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.22420
UR - https://arxiv.org/abs/2609.22420
AB - The many-body bootstrap certifies ground-state properties by minimizing energy over correlators constrained only by positivity, without any variational wavefunction. Its cost, however, grows exponentially with the size of the operators involved, placing the long-range string correlators that diagnose topological phases out of reach. We overcome this by treating string operators as primary objects: bare and endpoint-dressed strings satisfy a closed operator algebra with one another and with local words, yielding a semidefinite program whose cost grows only polynomially with string length. A single computation then yields estimates of string correlators of all lengths. Rigorous two-sided bounds are obtained for each target string once the ground-state energy is pinned within a window. When benchmarked against density-matrix renormalization group computations on the cluster Ising and spin−1 Heisenberg chains, the method certifies the nonlocal string order of the cluster and Haldane phases directly from the Hamiltonian. Our rigorous certification of string order parameters reveal that the sharpness of the bounds is set by which operators enter the calculation, not by their length alone. By exploiting algebraic closure of strings and local words, our framework lifts spatially extended observables into polynomially tractable bootstrap variables, opening up a route to wavefunction-free certification of nonlocal order in quantum many-body systems.
ER -