Ab initio derivation and exact diagonalization analysis of low-energy effective Hamiltonians for ${\beta }^{\prime }\text{−}\mathrm{X}{\left[\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2\right]}_2$

The molecular solids ${\beta }^{\prime }\text{−}\text{X}{\left[\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2\right]}_2$ (where X represents a cation) are typical compounds whose electronic structures are described by single-orbital Hubbard-type Hamiltonians with geometrical frustration. Using the ab initio downfolding method, we derive the low-energy effective Hamiltonians for ${\beta }^{\prime }\text{−}\text{X}{\left[\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2\right]}_2$ with available room- and low-temperature structures. We find that the amplitudes of the Coulomb interactions and the anisotropy of the hopping parameters in the effective Hamiltonians are sensitive to the changes in the lattice constants induced by lowering the temperature. The obtained effective Hamiltonians are analyzed using the exact diagonalization method with the boundary condition average. We find that a significant reduction of the antiferromagnetic ordered moment occurs in the effective Hamiltonian of ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}{\left[\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2\right]}_2$ with the low-temperature structure. The reduction is consistent with the quantum spin liquid behavior observed in experiments. The comprehensive derivations of the effective Hamiltonians and exact-diagonalization analyses of them will clarify the microscopic origins of the exotic quantum states of matter found in ${\beta }^{\prime }\text{−}\text{X}{\left[\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2\right]}_2$ such as the quantum spin liquid behavior.

Figure 1

Band dispersion of ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}{\left[\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2\right]}_2$ at (a) 5 K and (b) RT. The solid lines are obtained by DFT calculations, while the broken lines are obtained using the maximally localized Wannier functions (MLWFs). The energy is shifted so that the Fermi energy is zero. (c) MLWF of ${\beta }^{\prime }\text{−}{\mathrm{EtMe}}_3\mathrm{Sb}{\left[\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2\right]}_2$ at 5 K. Band dispersion of ${\beta }^{\prime }\text{−}{\mathrm{Me}}_4\mathrm{P}{\left[\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2\right]}_2$ at (d) 8 K and (e) RT, and (f) MLWF of ${\beta }^{\prime }\text{−}{\mathrm{Me}}_4\mathrm{P}{\left[\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2\right]}_2$ at 8 K. MLWFs are drawn by using vesta [42].

Figure 2

(a) Lattice constants, (b) transfer integrals, and (c) Coulomb interactions for quasi-two-dimensional organic molecules ${\beta }^{\prime }\text{−}\mathrm{X}{\left[\mathrm{Pd}{\left(\mathrm{dmit}\right)}_2\right]}_2$ (monovalent cations X = ${\mathrm{Me}}_4\mathrm{Y}, {\mathrm{EtMe}}_3\mathrm{Y}$, and ${\mathrm{Et}}_2{\mathrm{Me}}_2\mathrm{Y}$, where the pnictogen Y is P, As, or Sb). The open and closed symbols represent the data obtained using x-ray structures at RT and LT, respectively. (d) Compound dependence of $U/t_a, (U-{\mathrm{\Delta }}_{\mathrm{DDF}})/t_a$, and $(t_c-t_b)/t_a$. The inset shows the schematic of the lattice structure. The closed circles indicate MLWFs and each bold line in the closed circles shows a dmit molecule.

Figure 3

Boundary-condition dependence of the peak value of the spin structure factors $[S({\mathit{q}}_{\mathrm{peak}},\mathit{\varphi })/N_{\mathrm{s}}]$ for the effective Hamiltonians of (a) ${\mathrm{Me}}_4\mathrm{P}$ with an LT structure and (b) ${\mathrm{EtMe}}_3\mathrm{Sb}$ with an LT structure. For ${\mathrm{Me}}_4\mathrm{P}, {\mathit{q}}_{\mathrm{peak}}=(\pi ,0)$ for all the boundary conditions. Although ${\mathit{q}}_{\mathrm{peak}}=(\pi ,0)$ for ${\mathrm{EtMe}}_3\mathrm{Sb}$ for most boundary conditions, ${\mathit{q}}_{\mathrm{peak}}=(\pi ,\pi /2)$ in 30 boundary conditions. As we explained in Sec. II B, we calculate $-\pi \le {\varphi }_x<\pi$ and $0\le {\varphi }_y\le \pi$ with division width $\pi /8$. Using the relations $S({\mathit{q}}_{\mathrm{peak}},\mathit{\varphi })=S({\mathit{q}}_{\mathrm{peak}},-\mathit{\varphi })$ and $S({\mathit{q}}_{\mathrm{peak}},({\varphi }_x=\pi ,{\varphi }_y))=S({\mathit{q}}_{\mathrm{peak}},({\varphi }_x=-\pi ,{\varphi }_y))$, we plot the peak values in the whole Brillouin zone.

Figure 4

Boundary-condition dependence of the chemical potentials ${\mu }^{+}\left(\mathit{\varphi }\right)$ and ${\mu }^{-}\left(\mathit{\varphi }\right)$ for the effective Hamiltonians of (a) ${\mathrm{Me}}_4\mathrm{P}$ salt with an LT structure and (b) ${\mathrm{EtMe}}_3\mathrm{Sb}$ salt with an LT structure. For clarity, the energy is shifted so that ${min}_{\mathit{\varphi }}{\mu }^{-}\left(\mathit{\varphi }\right)$ is zero. In the bottom panel, ${\mu }^{+}\left(\mathit{\varphi }\right)-{\mu }^{-}\left(\mathit{\varphi }\right)$ is shown.

Figure 5

(a) Compound dependence of the peak values of the boundary averaged spin structure factors $S\left({\mathit{q}}_{\mathrm{peak}}\right)/N_{\mathrm{s}}$ as a function of $(t_c-t_b)/t_a$. For $(t_c-t_b)/t_a\ll 0$, the stripe magnetic ordered phase becomes stable, while the Néel-type magnetic ordered phase becomes stable for $(t_c-t_b)/t_a\gg 0$. Sandwiched by two magnetic ordered phases, around $(t_c-t_b)/t_a=0$, the spin structure factors are significantly reduced. The error bars are estimated as the standard errors of the boundary average (see main text). (b) Compound dependence of the charge gap as a function of $(U-{\mathrm{\Delta }}_{\mathrm{DDF}})/t_a$. The on-site-Coulomb dependence of the charge gap is fitted by the function $\alpha (U/t_a-U_c/t_a)$, where $\alpha =0.84\left(3\right)$ and $U_c/t_a=5.1\left(2\right)$. The broken line is the result of the fitting.

Figure 6

Boundary-averaged spin structure factors for RT structures (upper panel) and LT structures (lower panel).