# P3668: Exact ground energy of the antiferromagnetic Heisenberg model on the Petersen graph

- ID: `P3668`
- Reference: `petersen-antiferromagnetic-heisenberg-ground-polynomial`
- Page: https://theoremdb.org/statements/P3668
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/petersen-antiferromagnetic-heisenberg-ground-polynomial?representation=markdown
- Record maturity: Reviewed problem

## The problem

For the Petersen graph \(G\), determine the exact smallest eigenvalue of \(H=\sum_{uv\in E(G)}(X_uX_v+Y_uY_v+Z_uZ_v)\) on ten spin-one-half sites, including its minimal polynomial over \(\mathbb Q\).

### Remarks

- **Remark.** X, Y, and Z are the Pauli matrices acting on the indicated tensor factor.
- **Remark.** The Hamiltonian acts on (C^2) tensor power 10 and preserves total Z-spin.

### What counts as a solution

- Give the irreducible minimal polynomial, a rational isolating interval for its ground-state root, and exact block factorizations proving global minimality.

## Status

**Resolution status:** Open.

No proof or formalization has been recorded.

## Research packet

### Background and intake notes

The graph has enough symmetry for representation-theoretic compression, while the unreduced block is still large enough to make exact characteristic-polynomial work substantial.

- Work in fixed magnetization sectors and then decompose under the Petersen automorphism group. The zero-magnetization block has dimension C(10,5)=252.
- A decimal eigenvalue is only a lead. Certification needs an exact factor of the integer characteristic polynomial, an isolating interval, and proof that all other blocks lie above it.

- Recorded example: With outer-cycle vertices 0 through 4, inner-star vertices 5 through 9, and spokes i to 5+i, the Hamiltonian has fifteen edge terms.

### Computational notes

- Numerical diagonalization of the 252 by 252 zero-magnetization block gave lowest eigenvalue -17.357816691600558. The next distinct numerical eigenvalue was -17.108523423195486.
- The matrix was assembled with integer diagonal entries plus off-diagonal spin-exchange entries equal to 2. The lowest numerical value appeared with multiplicity five inside this block; exact multiplicity remains to be certified.

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `petersen-antiferromagnetic-heisenberg-ground-polynomial`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.