CM01 — Hubbard Model Mott Transition
| Field | Value |
|---|---|
| Domain | Condensed Matter |
| System | Hubbard model at half-filling |
| Group | SU(2) (spin rotation) |
| H^k tier | H¹ |
| ISA | Forge (β ≈ β*) |
| Status | Validated |
| Opcodes | ORBIT · TWIST · BIND · LABEL |
| Papers | Paper 563 |
Physical system
The Hubbard model is the minimal model of interacting electrons on a lattice:
H = −t Σ_{⟨ij⟩,σ} c†{iσ} c{jσ} + U Σ_i n_{i↑} n_{i↓}
The hopping term t allows electrons to delocalise (ORBIT across the lattice); the on-site repulsion U penalises double occupancy (two electrons on the same site). At half-filling (one electron per site on average), the competition between t and U drives the Mott metal-insulator transition at a critical ratio U_c/t:
- U/t ≪ 1 (metallic): electrons delocalise freely; Fermi liquid; ORBIT dominates; band theory applies.
- U/t ≈ U_c/t ≈ 1.8 (critical, β* snap): double occupancy collapses; spectral weight transfers from quasiparticle peak to Hubbard bands; the Fermi surface topology changes.
- U/t ≫ 1 (Mott insulator): electrons localise; one per site; charge gap 2U opens; spin physics described by Heisenberg antiferromagnet with J = 4t²/U; BIND dominates (superexchange = ring exchange = H² holonomy).
x563c experiment (Paper 563, SHA 174927e): Galerkin solver confirmed the Mott β* snap at U/t ≈ 1.8 via double-occupancy collapse — directly validating the ISA snap condition for the Hubbard model.
Target category
FermiLat(N) — the category of fermionic lattice models on N sites with SU(2) spin symmetry. Objects: N-site Fock spaces ⊗ᵢ (|0⟩, |↑⟩, |↓⟩, |↑↓⟩). Morphisms: particle-hole symmetric unitaries preserving the half-filling constraint ⟨n⟩ = 1. The Mott insulator is the terminal object in the U/t → ∞ limit: a product state of localised spins with no charge fluctuations.
Interpretation functor
F: C → FermiLat(N) defined by:
| Opcode | F(opcode) |
|---|---|
| ORBIT | Hopping: c†{jσ} c{iσ} moves an electron from site i to site j; the kinetic ORBIT on the lattice; generates the Fermi sea in the non-interacting limit |
| TWIST | Exchange: virtual hopping c†{iσ} c{jσ} c†{jσ’} c{iσ’} creates a Berry phase in the spin sector; generates the Heisenberg exchange J = 4t²/U in the Mott limit; the H¹ spin correlation |
| BIND | Superexchange ring: four-site ring exchange t⁴/U³ term; generates the four-spin BIND interaction; becomes important near the Mott transition where ring exchange corrections to Heisenberg model are O(t/U)² |
| LABEL | Double occupancy D = ⟨n_{i↑} n_{i↓}⟩: the order parameter; D ≈ 0.25 (free) → D ≈ 0 (Mott); the β* snap is D collapsing from ~0.15 to ~0.02 at U_c/t ≈ 1.8 (x563c) |
ISA programme
INIT: LABEL[half-filling: <n> = 1 per site] -- constraint
FREE: ORBIT[t * c†_j c_i | hopping, U=0 limit] -- Fermi sea (H0)
INTERACT: LABEL[U * n_up * n_dn | on-site repulsion] -- H0 energy penalty
DOUBLOCC: LABEL[D = <n_up n_dn> | double occupancy] -- order parameter
SNAP?: LABEL[D collapsing? | U/t near 1.8] -- beta* test (x563c)
EXCHANGE: TWIST[J = 4t^2/U | spin exchange in Mott limit] -- H1 spin physics
RING: BIND[K = 4t^4/U^3 | ring exchange] -- H2 correction
OUTPUT: LABEL[metal if U<Uc, Mott insulator if U>Uc] -- phase label
Computable output
- Double occupancy D(U/t): computed by x563c (Galerkin solver) for the 1D Hubbard chain. D decreases monotonically from D = 0.25 (U=0, free fermions) through D ≈ 0.15 (U/t = 1) to D ≈ 0.02 (U/t = 4). The β* snap at U/t ≈ 1.8 is where dD/d(U/t) is maximum — the sharpest change in double occupancy, the ISA snap condition. Validated against exact Bethe ansatz results (Lieb-Wu 1968) for 1D; DMFT for infinite dimensions.
- Mott gap Δ = U − W (U large): charge gap that opens in the insulating phase. For U ≫ t: Δ ≈ U − 2zt where z is the coordination number. The two Hubbard bands (lower: singly-occupied; upper: doubly-occupied) are LABEL eigenvalues separated by Δ. In 1D (Bethe ansatz): Δ = 0 for all U > 0 (no true Mott transition in 1D at T=0), but D still shows the crossover.
- Heisenberg exchange J = 4t²/U in the Mott limit: the TWIST output. Néel temperature T_N ∝ J; spin-wave velocity v_s = J√2 (square lattice). Confirmed in cuprate parent compounds (La₂CuO₄: J ≈ 130 meV, U/t ≈ 8).
- Spectral weight transfer: in ARPES, the quasiparticle peak at the Fermi energy loses spectral weight Z as U increases; Z → 0 at the Mott transition (Brinkman-Rice theory). Z is the ORBIT amplitude — it measures how much of the electron propagation is coherent (ORBIT-like) vs incoherent (blocked by U).
The Mott transition as β* snap
The Mott transition is the archetype of a correlation-driven β* snap: it is not driven by symmetry breaking (no order parameter in the Landau sense for the charge sector at half-filling in the paramagnetic Mott state) but by the change in the ORBIT topology of the Fermi surface.
In the ISA framework:
- Metallic phase (U < U_c): the Fermi surface is a large ORBIT enclosing half the Brillouin zone (Luttinger theorem). ORBIT is the dominant opcode; TWIST gives spin fluctuations; BIND is perturbative.
- At U_c (β* snap): the quasiparticle residue Z → 0; the ORBIT coherence collapses; spectral weight transfers from quasiparticle peak to incoherent Hubbard bands. The Fermi surface topology changes from a large electron Fermi surface to (in the Mott state) no Fermi surface at all.
- Mott insulating phase (U > U_c): no ORBIT (charge is localised); TWIST dominates (spin exchange J = 4t²/U via virtual hopping); BIND appears as ring exchange corrections K = 4t⁴/U³.
The snap condition from x563c: double occupancy D collapses at U/t ≈ 1.8 for the Hubbard chain. This is the ISA snap: D is the ORBIT overlap (probability of two electrons sharing a site), and it measures the coherence of the hopping ORBIT. When D → 0, ORBIT is suppressed and TWIST takes over.
Connection to cuprate superconductivity
The cuprate high-temperature superconductors (La₂CuO₄, YBa₂Cu₃O₇, …) are doped Mott insulators — they sit at U/t just above the Mott transition, doped with holes that can move through the localised spin background. The ISA story:
| Doping level | Phase | ISA |
|---|---|---|
| 0 (undoped) | Mott antiferromagnet | TWIST (Heisenberg J) |
| Under-doped | Pseudogap | TWIST + partial ORBIT (doped holes) |
| Optimal doped | d-wave superconductor | BIND (Cooper pairs via spin fluctuation exchange) |
| Over-doped | Conventional metal | ORBIT (Fermi liquid recovers) |
The superconducting dome is the region where BIND (Cooper pairing from TWIST spin fluctuations) wins over ORBIT (Fermi liquid). The pseudogap is the TWIST- dominated regime where spin correlations persist but charge order is absent. The Mott transition at U_c is the β* snap that makes all of this possible — without it, cuprates would be ordinary metals.
Cold-atom validation
The Hubbard model has been realised exactly in ultracold fermionic atoms (⁴⁰K or ⁶Li) in optical lattices, where U/t is tunable via Feshbach resonances and lattice depth:
- Jördens et al. (2008), Nature 455, 204: Mott insulating state observed at half-filling via double-occupancy suppression — directly measuring D ≈ 0 in the Mott phase. The ISA LABEL output (D → 0) confirmed.
- Greif et al. (2013), Science 340, 1307: short-range spin correlations (TWIST, Heisenberg exchange J) measured via spin-sensitive imaging.
- Mazurenko et al. (2017), Nature 545, 462: long-range antiferromagnetic order (TWIST ordering) imaged directly at U/t ≈ 7.
Validation
- Lieb & Wu (1968): exact Bethe ansatz solution for 1D Hubbard chain; no true Mott gap in 1D at T=0, but D(U/t) exactly computed.
- x563c (Paper 563, SHA 174927e): Galerkin solver; β* snap at U/t ≈ 1.8 confirmed numerically.
- DMFT (Georges et al. 1996, Rev. Mod. Phys.): exact solution in d=∞; Mott transition at U_c/t = 2.9√2; Z → 0 confirmed analytically.
- Cuprates: J ≈ 130 meV in La₂CuO₄ from neutron scattering (Coldea et al. 2001); consistent with J = 4t²/U at U/t ≈ 8.