Sharp Bounds on Ground State Energy of the SYK Model

Abstract

We study the Sachdev-Ye-Kitaev (SYK) Hamiltonian H_SYK on n Majorana modes with k-body interactions, and prove that 𝔼|H_SYK|_op = (1 − o(1))·√(2n)/k for super-constant k≤ o(√n), where the expectation is over the disorder variables in the Hamiltonian. This confirms predictions due to Garcia-Garcia, Jia and Verbaarschot and answers a question posed by Feng, Tian, and Wei. Our results extend to the sparse SYK Hamiltonian. As a corollary, we obtain that a dissipative quantum algorithm due to Basso, Chen, and Dalzell provably computes the ground state energy of the SYK Hamiltonian up to an O(1)-multiplicative factor for all k < √n/4. Our key technical idea is identifying an explicit, deterministic linear operator x such that a fixed quadratic form of x^2ℓ exactly equals the expected trace moments of the SYK Hamiltonian for every n and k. This linear operator can be naturally viewed as a twisted model of bosons on the space of hyperedges of a hypergraph. The problem thus reduces to identifying the spectral edge of x, which we show is dominated by the spectrum of a natural n choose k-dimensional matrix from the Johnson scheme and is straightforward to compute using known results. To show that our bound is sharp, we construct a witness state with a large quadratic form on x and transform it into a certificate of a lower bound on the largest quadratic form on H_SYK. Beyond the SYK Hamiltonian, our techniques also apply to the k−local quantum spin glass (QSG), yielding 𝔼|H_k−QSG|_op≤ O(√(n/k)) for all 1≤ k≤√n/2, thus improving the standard O(√n) bound by a factor of √k.

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BibTeX RIS

BibTeX

@misc{basu2026sharp,
  title = {{Sharp Bounds on Ground State Energy of the SYK Model}},
  author = {Arpon Basu and Pravesh K. Kothari and Siddhant Midha},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.27185},
  eprint = {2607.27185},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.27185},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Sharp Bounds on Ground State Energy of the SYK Model
AU  - Arpon Basu
AU  - Pravesh K. Kothari
AU  - Siddhant Midha
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2607.27185
UR  - https://arxiv.org/abs/2607.27185
AB  - We study the Sachdev-Ye-Kitaev (SYK) Hamiltonian H_SYK on n Majorana modes with k-body interactions, and prove that 𝔼|H_SYK|_op = (1 − o(1))·√(2n)/k for super-constant k≤ o(√n), where the expectation is over the disorder variables in the Hamiltonian. This confirms predictions due to Garcia-Garcia, Jia and Verbaarschot and answers a question posed by Feng, Tian, and Wei. Our results extend to the sparse SYK Hamiltonian. As a corollary, we obtain that a dissipative quantum algorithm due to Basso, Chen, and Dalzell provably computes the ground state energy of the SYK Hamiltonian up to an O(1)-multiplicative factor for all k < √n/4. Our key technical idea is identifying an explicit, deterministic linear operator x such that a fixed quadratic form of x^2ℓ exactly equals the expected trace moments of the SYK Hamiltonian for every n and k. This linear operator can be naturally viewed as a twisted model of bosons on the space of hyperedges of a hypergraph. The problem thus reduces to identifying the spectral edge of x, which we show is dominated by the spectrum of a natural n choose k-dimensional matrix from the Johnson scheme and is straightforward to compute using known results. To show that our bound is sharp, we construct a witness state with a large quadratic form on x and transform it into a certificate of a lower bound on the largest quadratic form on H_SYK. Beyond the SYK Hamiltonian, our techniques also apply to the k−local quantum spin glass (QSG), yielding 𝔼|H_k−QSG|_op≤ O(√(n/k)) for all 1≤ k≤√n/2, thus improving the standard O(√n) bound by a factor of √k.
ER  -

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