Symbol

Meaning

Def. Equation

${\widehat{H}}_{sq}$

Hamiltonian on the square lattice

(1)

${\widehat{H}}_{sq\mathrm{,}g}$

Hamiltonian on the square lattice with ${\widehat{V}}_g$

(5)

${\widehat{H}}_{sq\mathrm{,}g\mathrm{,}DS}$

Hamiltonian for the degenerate states

(7)

${\widehat{H}}_{sq\mathrm{,}g\mathrm{,}NG}$

Hamiltonian for Nambu-Goldstone mode

(9)

${\widehat{H}}_{sq\mathrm{,}g\mathrm{,}ex}$

Whole Hamiltonian of the extended system with ${\widehat{V}}_g$

(11)

${\widehat{H}}_{eff\mathrm{,}ex}$

Effective Hamiltonian on the extended system

(12)

${\widehat{H}}_{CW}$

Hamiltonian of Curie-Weiss model

(19)

${\widehat{H}}_{small}$

Hamiltonian on the16+8 lattice

${\widehat{H}}_{eff\mathrm{,}a\mathrm{,}small}$

Effective Hamiltonian on the16+8 lattice

(31)

${\widehat{V}}_g$

Interaction of the explicit symmetry breaking

(4)

${\widehat{V}}_{ex}$

Additional interaction for the extended system

(10)

$E_0\left(N_{sq}\right)$

Ground state energy of ${\widehat{H}}_{sq}$ for the degenerate states

(7)

$E_{Gl}$

l-th energy of ${\widehat{H}}_{sq\mathrm{,}g\mathrm{,}DS}$

(50)

$E_{NG}$

Ground state energy of ${\widehat{H}}_{sq\mathrm{,}g\mathrm{,}NG}$

(9)

$E_{CW}\left(J_{CW}\right)$

Energy of Curie-Weiss model

(20)

$E_{ex}$

Energy by the high temperature expansion

(27)

$E_{eff\mathrm{,}a\mathrm{,}small}\left(J\mathrm{,}{J}_z\right)$

Energy of ${\widehat{H}}_{eff\mathrm{,}a\mathrm{,}small}$

(32)

$E_{cal}\left(u\mathrm{,}{J}_z\right)$

Energy calculated on the small lattice

$N_{sq}$

Size of the square lattice

$N_{ex}$

Size of the extended system

$N_{CW}$

Size of Curie-Weiss model

g

Strength of ${\widehat{V}}_g$

u

Strength of ${\widehat{V}}_{ex}$

${\widehat{S}}_i^{\alpha }$

Spin operator on the square lattice

${\widehat{S}}_{a\left(i\right)}^{\alpha }$

Spin operator in the extended system

${\widehat{S}}_a^{\alpha }$

Spin operator in Curie-Weiss model

$c\left(\alpha \mathrm{,}i\right)$

Coefficient in the first order of u in ${\widehat{H}}_{eff\mathrm{,}ex}$

(12)

$c\left(\alpha \mathrm{,}\beta \mathrm{,}i\mathrm{,}j\right)$

Coefficient in the second order of u in ${\widehat{H}}_{eff\mathrm{,}ex}$

(12)

$c_{DS}\left(\alpha \mathrm{,}\beta \mathrm{,}i\mathrm{,}j\right)$

Second order coefficient due to the degenerate states

(13)

$c_{NG}\left(\alpha \mathrm{,}\beta \mathrm{,}i\mathrm{,}j\right)$

Second order coefficient due to Nambu-Goldstone mode

(13)