Fock space and symmetry sectors

Let \(p=0,\ldots,M-1\) label an ordered set of one-particle modes. A Fock state is specified by its occupations,

\[ |\mathbf n\rangle=|n_0,n_1,\ldots,n_{M-1}\rangle, \]

and a many-body state in a finite sector is

\[ |\psi\rangle=\sum_{\alpha=1}^{D} A_\alpha|\alpha\rangle, \qquad \sum_\alpha |A_\alpha|^2=1. \]

The basis states \(|\alpha\rangle\) are occupation configurations consistent with the particle statistics and the conserved quantum numbers chosen for the calculation.

Fermions

For fermions,

\[ n_p\in\{0,1\}, \]

and the canonical anticommutation relations are

\[ \{c_p,c_q^\dagger\}=\delta_{pq}, \qquad \{c_p,c_q\}=\{c_p^\dagger,c_q^\dagger\}=0. \]

The total number operator is

\[ \hat N=\sum_{p=0}^{M-1} n_p, \qquad n_p=c_p^\dagger c_p. \]

A fixed-\(N\) sector contains

\[ D_F(M,N)=\binom{M}{N} \]

basis states.

Bosons

For bosons,

\[ n_p=0,1,2,\ldots, \]

with

\[ [b_p,b_q^\dagger]=\delta_{pq}, \qquad [b_p,b_q]=[b_p^\dagger,b_q^\dagger]=0. \]

At fixed total particle number,

\[ \sum_{p=0}^{M-1}n_p=N, \]

so the sector dimension is the number of weak compositions of \(N\) into \(M\) parts,

\[ D_B(M,N)=\binom{N+M-1}{N}. \]

A fixed-\(N\) bosonic sector is finite without imposing an additional local occupation cutoff.

Conserved quantum numbers

If a Hermitian operator \(Q\) commutes with the Hamiltonian,

\[ [H,Q]=0, \]

then \(H\) can be block diagonalized in eigenspaces of \(Q\). Particle number is the most common example. If several commuting number operators are conserved, the basis may be restricted to their simultaneous eigenspace.

For a spin-conserving fermion model,

\[ \hat N_\uparrow=\sum_i n_{i\uparrow}, \qquad \hat N_\downarrow=\sum_i n_{i\downarrow}, \]

and a sector with fixed \((N_\uparrow,N_\downarrow)\) has

\[ D=\binom{L}{N_\uparrow}\binom{L}{N_\downarrow} \]

states on \(L\) sites. Analogous constraints apply to separately conserved layers, orbitals, species, or other labeled subspaces. These are symmetry-constrained sectors: the constraint is valid only when its number operator commutes with the Hamiltonian.

Matrix elements

Once a basis has been chosen, the Hamiltonian is the finite matrix

\[ H_{\alpha\beta}=\langle\alpha|H|\beta\rangle. \]

The exact-diagonalization problem is therefore exact for the represented finite Hamiltonian up to floating-point error and eigensolver tolerance. Its practical limitation is combinatorial growth of \(D\), even after symmetry reduction. The scaling of representative fermionic and bosonic sectors is shown in Fock space, modes, and symmetry sectors.