Table 2 Quantum numbers and overlap on the torus.
From: Local models of fractional quantum Hall states in lattices and physical implementation
N | L x × L y |
| f | t x | t y | D | |〈ψT0| |
|
| f | t x | t y | D | |〈ψT1| |
|
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
12 | 4 × 3 | 0 | +1 | 0 | 0 | 44 | 0.9509 | 0.9958 | 0 | +1 | 2 | 0 | 42 | 0.9765 | 0.9980 |
16 | 4 × 4 | 0 | +1 | 0 | 0 | 441 | 0.9697 | 0.9981 | 0 | +1 | 0 | 0 | 441 | 0.9244 | 0.9951 |
20 | 4 × 5 | 0 | +1 | 0 | 0 | 4,654 | 0.9581 | 0.9979 | 0 | +1 | 2 | 0 | 4,650 | 0.9717 | 0.9986 |
30 | 6 × 5 | 0 | −1 | 3 | 0 | 2,585,850 | 0.9423 | 0.9980 | 0 | −1 | 0 | 0 | 2,584,754 | 0.9697 | 0.9990 |
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- The table provides the overlap and overlap per site between the two lowest energy eigenstates, ψT0 and ψT1, of the Hamiltonian in equation (3) with φ1=0.07 × 2π, φ2=0.03 × 2π and periodic boundary conditions and the CFT states,
and 
, in equation (4). 
is the eigenvalue of the z component of the total spin of the states, f is the eigenvalue of the operator that flips all the spins, 
is the eigenvalue of the translation operator by one lattice constant in the x direction (y direction) and D is the dimension of the subspace of Hilbert space that consists of all states with the given eigenvalues (columns 3–7 are for ψT0 and columns 10–14 are for ψT1. In general, 
=0 and f=(−1)N/2 for the CFT states. For even-by-even lattices tx=ty=0 for both CFT states and for even-by-odd lattices 
and ty=0 for 
and 
and ty=0 for 
.
and 
, in equation (4). 
is the eigenvalue of the z component of the total spin of the states, f is the eigenvalue of the operator that flips all the spins, 
is the eigenvalue of the translation operator by one lattice constant in the x direction (y direction) and D is the dimension of the subspace of Hilbert space that consists of all states with the given eigenvalues (columns 3–7 are for ψT0 and columns 10–14 are for ψT1. In general, 
=0 and f=(−1)N/2 for the CFT states. For even-by-even lattices tx=ty=0 for both CFT states and for even-by-odd lattices 
and ty=0 for 
and 
and ty=0 for 
.