Math 2.0

Statistical mechanics, structure of matter Quantum spin chains and lattice fermions Theorem

Nagaoka ferromagnetism

Builds on Hubbard model

Statement

Let Λ\Lambda be a finite set of sites. Consider the Hubbard model H=∑x,y∈Λ∑σtxy cxσ†cyσ+U∑x∈Λnx↑nx↓H = \sum_{x, y \in \Lambda} \sum_{\sigma} t_{xy}\, c_{x\sigma}^\dagger c_{y\sigma} + U \sum_{x \in \Lambda} n_{x\uparrow} n_{x\downarrow} with real symmetric hopping txy=tyx≥0t_{xy} = t_{yx} \ge 0. Take the limit U=∞U = \infty, that is, restrict to states with no doubly occupied site. Let the number of electrons be ∣Λ∣−1\abs{\Lambda} - 1, so that there is exactly one hole. Assume the connectivity condition: any two spin configurations with one hole are joined by some sequence of hops of the hole along bonds with txy>0t_{xy} > 0. Then the ground states have the maximal total spin Smax⁡=(∣Λ∣−1)/2S_{\max} = (\abs{\Lambda} - 1)/2, and they form a single spin multiplet, of dimension 2Smax⁡+12S_{\max} + 1.

Remarks

The proof uses a Perron–Frobenius argument. In a suitable basis of configurations of the hole and the spins, the off-diagonal matrix elements of −H-H are non-negative, thanks to the sign condition on txyt_{xy}. The connectivity condition makes this matrix irreducible within each sector of fixed StotzS^z_{\mathrm{tot}}. So the ground state in each sector is unique and has positive coefficients, and comparing with the fully polarized state shows that it has maximal total spin. Nagaoka proved the result in 1966, and Thouless gave a related argument independently; Tasaki gave the general version above. The square and simple cubic lattices with nearest-neighbour hopping satisfy the connectivity condition. They are bipartite, so the sign of the hopping can be reversed by a gauge transformation, and the sign condition is no restriction there. A one-dimensional open chain does not satisfy the connectivity condition, since the hole cannot change the order of the spins. With two or more holes, or at finite UU, the ground state is not known in general.

Going deeper

For a student

Nagaoka's theorem is a rare exact result about the Hubbard model away from half filling. It shows that kinetic energy, not exchange, can drive ferromagnetism, a mechanism that is very different from the Heisenberg picture. Its hypotheses are delicate: the sign of the hopping matters on non-bipartite lattices, and a single hole is a vanishing density. The stability of the ferromagnet at a finite density of holes is an open problem. Related rigorous results on flat-band ferromagnetism, by Mielke and Tasaki, give other routes to ferromagnetism in Hubbard models.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Almost Mathieu operator, example, no evidence levelAlmost Mathieu operat…exampleBethe ansatz, concept, no evidence levelBethe ansatzconceptBrillouin zone, concept, no evidence levelBrillouin zoneconceptLattice model, concept, no evidence levelLattice modelconceptSelf-adjoint operator, concept, no evidence levelSelf-adjoint operatorconceptTensor product of Hilbert spaces, concept, no evidence levelTensor product of Hil…conceptTight-binding model, example, no evidence levelTight-binding modelexampleQuantum Heisenberg model, example, no evidence levelQuantum Heisenberg mo…exampleHubbard model, example, no evidence levelHubbard modelexampleNagaoka ferromagnetism, statement, no evidence levelNagaoka ferromagnetismstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Almost Mathieu operator example
  2. Bethe ansatz concept
  3. Brillouin zone concept
  4. Lattice model concept
  5. Self-adjoint operator concept
  6. Tensor product of Hilbert spaces concept
  7. Tight-binding model example
  8. Quantum Heisenberg model example
  9. Hubbard model example
  10. Nagaoka ferromagnetism statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Hubbard_model
About this page
TagRVB2FA29DD
URIm2://RVB2FA29DD
Kindstatement, theorem
WritingW0 stub
MSC82B20, 82D40, 82B10
StatusThis copy of the site has no build data. Run m2 status RVB2FA29DD.
AboutHubbard model
UsesHubbard model
Changed3 November 2026

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