Periodic Hubbard filling, spin sectors, and attractive U¶

Hubbard family · Supported models · Hubbard equations and numerical method

Use --particles N and --sz VALUE to select the physical sector. The two spin populations are \(N_{\mathrm{up}} =N /2+S^z\) and \(N_{\mathrm{down}} =N /2-S^z\); each must be an integer between zero and L. Half-integers can be written as 1/2 or 0.5. The defaults remain N=L and Sz=0, and L must be even. This guide describes bethe-hubbard-pbc. The free-end solver uses the same energy mappings without the periodic shell-parity restrictions; it also accepts odd L and defaults to Sz=1/2 when N is odd.

Supported ground states¶

The numerical core solves a repulsive sector with N<=L and N_down<=N_up. Exact symmetries reduce other physical sectors to this core.

Physical request	Current support
U=0	Every valid N and Sz; independent free-fermion bands
Only one spin species	Every valid N and either sign of U; no double occupation
U>0, N=L	Every valid Sz
U>0, N<L	Odd N_up and odd N_down, plus the single-species case
U>0, N>L	Full particle-hole image of a supported sector below half filling
U<0, Sz=0	Every even N from 0 to 2L, via a repulsive half-filled spin sector
U<0, Sz!=0	Supported when the mapped repulsive sector belongs to the families above

For example:

# Repulsive doping: three electrons of each spin.
bethe-hubbard-pbc 16 --u 4 --particles 6
# Electron doping: maps to six particles below half filling.
bethe-hubbard-pbc 16 --u 4 --particles 26
# Balanced attraction: four pairs, mapped to polarized half filling.
bethe-hubbard-pbc 16 --u -4 --particles 8 --roots
# A half-filled polarized repulsive sector.
bethe-hubbard-pbc 16 --u 4 --sz -2
# Odd N is allowed on the unrestricted free-fermion path.
bethe-hubbard-pbc 16 --u 0 --particles 7 --sz 1/2

This is deliberately not yet an arbitrary-filling, arbitrary-spin ground-state solver. In particular, 16 --u 4 --particles 8 --sz 0 is rejected: both spin populations are even. A periodic ring's shell parity can change which spin branch is lowest. Continuing a single compact quantum-number sea is not a general minimization over those branches or their spin descendants. Supporting these remaining sectors needs branch selection, not just more Newton variables. No complex-string or general excited-state solver is implied here.

Why attraction can use a repulsive solver¶

On an even bipartite ring, the down-spin particle-hole transformation

\[ c_{j,\downarrow}\longrightarrow(-1)^j c^\dagger_{j,\downarrow}. \]

preserves hopping and exchanges the sign of U. This is the Shiba mapping; see rylands-2022, Sec. II. For our unshifted Hamiltonian, writing g=|U|,

\[ \begin{aligned} (N_\uparrow,N_\downarrow)&\longrightarrow(N_\uparrow,L-N_\downarrow),\\{} E_{-g}(N_\uparrow,N_\downarrow)&=E_{+g}(N_\uparrow,L-N_\downarrow)-gN_\uparrow. \end{aligned} \]

Thus balanced attraction maps to N'=L with Sz'=(N-L)/2. The attractive energy can be found with real repulsive roots even though the physical attractive Bethe roots need not be real. This is an exact unitary mapping, not a string approximation. We return the auxiliary roots explicitly; we do not reconstruct the physical attractive roots or wavefunction.

Above half filling, a particle-hole transformation on both species gives

\[ \begin{aligned} (N_\uparrow,N_\downarrow)&\longrightarrow(L-N_\uparrow,L-N_\downarrow),\\{} E_U(N_\uparrow,N_\downarrow)&=E_U(L-N_\uparrow,L-N_\downarrow)+U(N-L). \end{aligned} \]

Finally, spin reversal exchanges N_up and N_down without changing energy. The implementation applies Shiba first when U<0, then full particle-hole if needed, then spin reversal. These operations preserve the minimum within the mapped sector. Odd rings are excluded because the staggered transformation would change the periodic boundary condition.

Momentum and root metadata¶

Momentum also transforms. In the convention P=sum(k) mod 2*pi, for even L, the down-spin Shiba step adds \(\pi \,(N_{\mathrm{down}} -1)\) when converting auxiliary momentum back to the physical sector. The full particle-hole step adds \(\pi \,N\). Counts in these expressions belong to the sector before that step; spin reversal adds nothing. These offsets follow by replacing each occupied down-spin momentum by its complementary hole at \(\pi -k\); a filled one-spin band has momentum pi. Tests check the resulting physical momenta directly against translation in Fock space, including degenerate ground eigenspaces.

In State<Real>:

particles, down_spins, interaction, energy, momentum_index, and momentum describe the requested physical sector.
root_particles, root_down_spins, and root_interaction describe charge_momenta, spin_rapidities, quantum_numbers, and all residuals.
energy_offset and momentum_offset obey E=E_roots+energy_offset and momentum_index=(root_index+momentum_offset)%L.
shiba_transformed, particle_hole_transformed, and spin_reversed record the mapping; auxiliary_roots() indicates whether any was used.

The CLI prints this distinction whenever a mapping was applied, including with --roots and for unconverged estimates. Free-fermion occupation lists have no Bethe labels or spin rapidities. For open shells, their deterministic representative fills q=-(count-1)/2,... with integer division toward zero; another degenerate representative may have a different momentum.

Both energy offsets and all solves use the selected scalar precision. No Uni20 linear-algebra extension is needed: the same real dense solve serves these sectors. Residuals are equation diagnostics, not energy-error bounds; in particular, adding an O(|U|N) attractive offset can conceal small binding or excitation scales at finite precision.