Table 1 One-body density matrix \({\rho }_{n{n}^{{\prime} }}\) among the essential W orbitals n in various stable ionic configurations

From: ‘Interaction annealing’ to determine effective quantized valence and orbital structure: an illustration with ferro-orbital order in WTe2

\({\rho }_{n{n}^{{\prime} }}\)

Td OP2

\({\rho }_{n{n}^{{\prime} }}\)

1T OP2

1T LS4

U

ag

eg1

eg2

U

ag

eg1

eg2

ag

eg1

eg2

 

0.54

−0.03

0.00

 

0.45

−0.01

0.00

0.46

−0.01

0.00

3

−0.03

0.38

0.00

8

−0.01

0.58

0.00

−0.01

0.57

0.00

 

0.00

0.00

0.57

 

0.00

0.00

0.53

0.00

0.00

0.53

 

0.20

0.03

0.00

 

0.12

−0.08

0.00

0.09

−0.00

0.00

20

0.03

0.18

0.00

20

−0.08

0.98

0.00

−0.00

0.81

0.00

 

0.00

0.00

0.96

 

0.00

0.00

0.15

0.00

0.00

0.80

  1. (Left column) For a realistic Td structure with intra-atomic interaction U = 3 eV, \({\rho }_{n{n}^{{\prime} }}\) contains fractional occupations (the diagonal elements) in all three orbitals due to strong charge fluctuation. Upon suppressing fluctuations through increased U = 20 eV, \({\rho }_{n{n}^{{\prime} }}\) shows clean occupation of only one of the orbitals, corresponding to an OP2 configuration of Fig. 2d. (Right columns) For a fictitious system of higher symmetric 1T structure and U = 8 eV, two stable configurations appear similar, but their interaction annealed counterpart reveals qualitatively distinct quantized ionic structures corresponding to an OP2 and a LS4 of Fig. 2