Fragment-orbital-dependent spin fluctuations in the single-component molecular conductor [Ni(dmdt)₂]

Dirac electron systems in solids are of interest to many researchers because of not only their quantum transport phenomena T.Ando1998 ; V.P.Gusynin2006 ; S.Murakami2004 ; D.Hsieh2008 and large diamagnetism, H.Fukuyama1970 ; H.Fukuyama2007 but also their unusual effects induced by the Coulomb interaction.A.A.Abrikosov1971 ; J.Gonzalez1994 ; J.Gonzalez1999 ; V.N.Kotov2012 ; T.O.Wehling2014 ; W.Witczak-Krempa2014

Dirac electron systems in molecular conductors, such as $\alpha$-(BEDT-TTF)₂I₃, provide suitable platforms for studying the effect of interaction because the electron hopping integrals between neighboring molecules are smaller than the on-site repulsive interactions reflecting the weak inter-molecular coupling. K.Kajita1992 ; N.Tajima2000 ; A.Kobayashi2004 ; S.Katayama2006 ; A.Kobayashi2007 ; M.O.Goerbig2008 ; K.Kajita2014 At high pressure, $\alpha$-(BEDT-TTF)₂I₃ is a massless Dirac electron system. However, at low pressure, a charge-ordered state appears presumably due to nearest-neighbor Coulomb repulsions, H.Seo2000 ; T.Takahashi2003 ; T.Takiuchi2007 where anomalous spin–charge separation on spin gaps K.Ishibashi2016 ; Y.Katayama2016 and transport phenomena occur.R.Beyer2016 ; D.Lu2016 ; D.Ohki2019 In addition, the long-range Coulomb interaction reshapes the Dirac cone because of a logarithmic velocity renormalization, which induces an anomalous magnetic response.M.Hirata2016 ; M.Hirata2017 ; A.Kobayashi2013 Moreover, ferrimagnetism and spin-triplet excitonic fluctuations are observed.D.Ohki2020 ; M.HirataRPP

The Dirac electron system in $\alpha$-(BEDT-TTF)₂I₃ is two-dimensional K.Kajita2014 because it is a layered molecular conductor and the hopping of electrons from one conducting layer to the neighboring one over the insulating anion layer is incoherent. By contrast, if the electron hopping perpendicular to the main conducting layer were coherent, the Dirac point would be connected and draw lines (rings) in the three-dimensional momentum space, which are called the Dirac nodal lines (rings). A.Burkov2011 ; C.-K.Chiu2014 ; C.Fang2015 ; Z.Gao2016

Such kinds of Dirac nodal line (ring) systems have indeed been found in graphite, P.R.Wallace1947 transition-metal monophosphates,H.Weng2015 Cu₃N,Y.Kim2015 antiperovskites,R.Yu2015 perovskite iridates,J.-M.Carter2012 and hexagonal pnictides with the composition CaAgX (X = P, As),A.Yamakage2016 as well as in the single-component molecular conductors [Pd(dddt)₂], R.Kato2017_A ; R.Kato2017_B ; Y.Suzumura2017_A ; Y.Suzumura2017_B ; Y.Suzumura2018_A ; Y.Suzumura2018_B ; T.Tsumuraya2018 ; Y.Suzumura2019 [Pt(dmdt)₂],B.Zhou2019 ; R.Kato2020 ; T.Kawamura2020 ; T.Kawamura2021 ; A.Kobayashi2021 and [Ni(dmdt)₂].T.Kawamura2021 ; A.Kobayashi2021

The Dirac nodal line (ring) systems exhibit not only the properties in common with two-dimensional Dirac electron systems, e.g., the in-plane conductivityB.Zhou2019 , but also the characteristic electronic properties such as non-dispersive Landau levels,J.-W.Rhim2015 Kondo effect,A.K.Mitchell2015 quasi-topological electromagnetic responses,S.T.Ramamurthy2017 and edge magnetismT.Kawamura2021 because of the three-dimensionality. However, the electron correlation effects on the Fermi surface in the Dirac nodal line systems have not yet been elucidated.

The prime focus of this study is such Dirac nodal line system in [M(dmdt)2] (M = Pt, Ni), which is a single-component molecular conductor that consists of the M(dmdt)₂ molecules, where the bracket [$\cdots$] stand for a crystal. This material is a triclinic system, as shown in Fig. 1, and has space-inversion symmetry. One unit cell contains one M(dmdt)₂ molecule. In previous studies, the electronic properties of [M(dmdt)₂] were studied using density functional theory (DFT), and tight-binding models were constructed on the basis of the extended Hückel method and DFT.B.Zhou2019 ; R.Kato2020 ; T.Kawamura2020 ; T.Kawamura2021 These investigations showed that [M(dmdt)₂] is a Dirac nodal line system. Furthermore, electronic resistivity measurements using conventional four-probe methods were performed and showed that the resistivity of [M(dmdt)₂] hardly depends on the temperature ($T$), which is consistent with the property of the Dirac electron system.B.Zhou2019 That is the universal conductivity.N.H.Shon1998 In addition, we previously suggested that the nesting between the Fermi arcs localized at the edge and the electronic correlation induce a helical spin density wave (SDW) at the edge.T.Kawamura2021

Recently, the spin-lattice relaxation rate $1/T_{1}T$, probing the low-energy spin dynamics, and the Knight shift, scaling to the spin susceptibility, of [Ni(dmdt)₂] were observed in a ¹³C nuclear magnetic resonance (¹³C-NMR) experiment.T.Sekine.Private At high temperature, $1/T_{1}T$ decreases with cooling and is almost proportional to $T^{2}$. However, at low temperature, it starts to increase with decreasing temperature and exhibits a peak structure at approximately $30$ K. Meanwhile, the Knight shift is almost proportional to $T$ because of the linear energy dispersion and does not increase. The mechanism of this anomalous temperature dependence of the spin fluctuations has not been elucidated.

In the present study, we theoretically investigate the electron correlation using the Fermi surface in [Ni(dmdt)₂] to elucidate the mechanism of this anomalous temperature dependence of the spin fluctuations. We calculate the spin susceptibility, Knight shift, and $1/T_{1}T$ using the random phase approximation (RPA) in a three-orbital Hubbard model describing [Ni(dmdt)₂], which is obtained using ab initio many-body perturbation theory calculations.

The electronic state of a molecular conductor is described by the molecular orbitals, which are linear combinations of the atomic orbitals in a molecule. The molecular orbital that has the highest energy and is fully occupied by electrons is called the highest occupied molecular orbital (HOMO), whereas the one having the lowest energy with no electrons is called the lowest unoccupied molecular orbital (LUMO). HOMO and LUMO are also called frontier orbitals. The electronic states of single-component molecular conductors, e.g., [M(tmdt)₂] (M = Ni, Au, Cu) and [M(dmdt)₂] (M = Pt, Ni), are described by multiple molecular orbitals localized in a part of the molecule.H.Seo2008 ; H.Seo2013 ; M.Tsuchiizu2012 ; T.Kawamura2020 ; T.Kawamura2021 These molecular orbitals are the energy eigenstates obtained using first-principles calculations and are called “fragment orbitals”. The fragment orbitals are transformed into HOMO and LUMO by a high-symmetry unitary transform.

Based on the band parameters determined from first-principle calculations, Seo et al. have constructed a Hubbard model of [M(tmdt)₂](M=Ni, Au, Cu), which is described by the fragment orbitals.H.Seo2008 ; H.Seo2013 The on-site repulsion acts between the same fragment orbitals that have spins of opposite signs, which is similar to the case of the present study. They have investigated the ordered state by calculating the electron density and spin density using mean-field approximation. By contrast, in this study, we will investigate the spin fluctuations by calculating the spin susceptibility on the basis of RPA. Furthermore, we show that the idea of the fragment orbitals is important for the physical properties of the Dirac nodal line systems in the single-component molecular conductor. As the other previous study of the fragment orbitals, some charge-transfer complexes such as (TTM-TTP)I₃ are also modeled by fragment orbitals.M.Tsuchiizu2011 ; M.Tsuchiizu2012

We found that the q=0 spin fluctuations are enhanced in two out of the three fragment orbitals, while an enhancement at an incommensurate wavenumber vector develops in the third orbital. Detailed analysis showed that these q=0 spin fluctuations do not correspond to a simple ferromagnetic fluctuations; rather, they are linked to an intra-molecular antiferromagnetic fluctuations. This implies that the spins of the fragment orbitals within the same molecule are inversely correlated. Further, q=0 implies a direct correlation of the spins between molecules. Using RPA, we determined that the $1/T_{1}T$ starts to increase at a low temperature by the q=0 spin fluctuations. By contrast, the Knight shift does not increase upon cooling because of the intra-molecular antiferromagnetic fluctuations.

At high temperature, an incommensurate spin fluctuations dominate the temperature dependence of $1/T_{1}T$. These magnetic responses are associated with the geometry of the Fermi surface and the characteristic wave functions of the $n$-orbital ($n\geq 3$) Dirac nodal line system. Thus, it is expected that other Dirac nodal line systems described by multiple-orbital models may have similar magnetic properties.

The remainder of this paper is organized as follows. In Section II, we introduce the spin susceptibility based on RPA and formulate $1/T_{1}T$ and the Knight shift. In Section III, we calculate the band structure, spin susceptibility, Knight shift, $1/T_{1}T$, and so on in the absence of interaction. In Section IV, we calculate the Stoner factor, Knight shift, and $1/T_{1}T$ in the presence of interaction by applying RPA to a Hubbard model. Section V draws conclusions.

We calculate the spin susceptibility, which incorporates the electron correlation effect within perturbation theory to investigate the enhancement of spin fluctuations in [Ni(dmdt)₂]. Furthermore, we calculate the Knight shift and the spin-lattice relaxation rate $1/T_{1}T$, which are the physical quantities observed using NMR. We apply RPA to the Hubbard model to calculate the spin susceptibility in [Ni(dmdt)₂]. Although calculations incorporating the self-energy, e.g., FLEX, are better approximations than RPA, RPA is more suitable for investigating spin fluctuations because the self-energy suppresses spin susceptibility.

The Hubbard Hamiltonian that we employ is given by

$$H=\sum_{\left<i,\alpha;j,\beta\right>,\sigma}t_{i,\alpha;j,\beta}c^{\dagger}_{i,\alpha,\sigma}c_{j,\beta,\sigma}+\sum_{i,\alpha}U_{\alpha}n_{i,\alpha,\uparrow}n_{i,\alpha,\downarrow},$$

(1)

where $i$ and $j$ are the unit-cell indices, and $\sigma$ is the spin index. Here, $t_{i,\alpha;j,\beta}$ is a transfer integral defined between the orbital $\alpha$ in the unit cell $i$ and the orbital $\beta$ in the cell $j$, and $U_{\alpha}$ represents the on-site repulsive interaction on the orbital $\alpha$, with $\alpha$ and $\beta$ standing for one of the three fragment orbitals in the unit cell (A, B, and C in Fig. 2). The indices $\alpha$ and $\beta$ represent the three fragment orbitals A, B, and C. $\sum_{\left<\cdots\right>}$ represents a summation that runs only for the hoppings that have a large energy scale than the cutoff (set to be $0.010$ eV in this study).

The electronic states of [Ni(dmdt)₂] near the Fermi energy $E_{F}$ are described by three fragment-decomposed Wannier orbitals(fragment orbitals) dubbed orbitals A, B, and C as illustrated in Fig. 2. They are obtained using Wannier fitting to three isolated energy bands near $E_{F}$, which were previously obtained using first-principles calculations. T.Kawamura2021

The Wannier fitting and first-principles calculations were performed using the programs respackK.Nakamura2020 and Quantum EspressoP.Giannozzi2017 , respectively.T.Kawamura2021 respack was also used for calculating the Coulomb interaction and other factors. Further, Quantum Espresso was used for first-principles calculations based on the pseudopotential method. Figure 2(a) and (b) show a side and vertical-axis view, respectively, of the molecule. The orbital B in Fig. 2(a) may look like a $d$ orbital, but is a $p$ orbital of the S atoms, as is evident from Fig. 2(b). The $d$ orbital of Ni is localized near the Ni atom, and its contribution to the orbital B is small. The orbitals A and C have $p$-orbital-like shapes and are equivalent because of space-inversion symmetry. In the present study, we assume that these three fragment orbitals sit in the same molecule and in the same unit cell.

By performing a Fourier transform, the Hamiltonian (Eq. 1) is rewritten as

$$\begin{split}H&=\sum_{\textbf{k},\alpha,\beta,\sigma}H^{0}_{\alpha\beta,\sigma}(\textbf{k})c^{\dagger}_{\textbf{k},\alpha,\sigma}c_{,\textbf{k},\beta,\sigma}\\ &+\frac{1}{N_{L}}\sum_{\textbf{k},\textbf{k}^{\prime},\textbf{q},\alpha}U_{\alpha\alpha}c^{\dagger}_{\textbf{k}+\textbf{q},\alpha,\uparrow}c^{\dagger}_{\textbf{k}^{\prime}-\textbf{q},\alpha,\downarrow}c_{\textbf{k}^{\prime},\alpha,\downarrow}c_{\textbf{k},\alpha,\uparrow},\end{split}$$

(2)

where k, $\textbf{k}^{\prime}$, and q are the wavenumber vectors. $N_{L}$ is the number of the unit cells in the system. The first term corresponds to the unperturbed Hamiltonian, and the second term is treated as a perturbed Hamiltonian. Here, $H^{0}_{\alpha\beta,\sigma}(\textbf{k})$ is defined as

$$H^{0}_{\alpha\beta,\sigma}(\textbf{k})=\sum_{\left<{\bm{\delta}}\right>}t_{\alpha\beta,\sigma,{\bm{\delta}}}e^{i\textbf{k}\cdot{\bm{\delta}}},$$

(3)

where ${\bm{\delta}}$ is a lattice vector connecting the neighbor unit cell. Further, $t_{\alpha\beta,\sigma,{\bm{\delta}}}$ is the transfer integral between the fragment orbitals $\alpha$ and $\beta$, which are separated by the lattice vector ${\bm{\delta}}$ and have the spin $\sigma$. We allot $t_{\alpha\beta,\sigma,{\bm{\delta}}}$ to the transfer integrals $t_{1}$, $t_{2}$, …, $t_{12}$ and the site potential $\Delta$ in Table 1, where $\Delta\equiv t_{BB,\sigma,\textbf{0}}-t_{AA,\sigma,\textbf{0}}$. Note that these transfer integrals were obtained from the Wannier fitting. And we omit the small hoppings whose sizes are less than a cutoff energy of $0.010$ eV to make the analysis simple (see Fig. 3 for the schematic illustration of such a hopping network). Note that $t_{1}$ connects the nearest-neighbor fragment orbitals within a molecule, while $t_{2}$, $t_{3}$, …, $t_{8}$ connect those between molecules in the crystalline $b$–$c$ plane, which corresponds to the $k_{b}$–$k_{c}$ plane in momentum space, where Dirac cones exist. We point out that $t_{1}$, $t_{2}$, and $t_{3}$ are the three essential transfer integrals that are needed to create the Dirac cones, while $t_{9}$ creates Fermi surfaces, and $t_{10}$, $t_{11}$, and $t_{12}$ wind the Dirac nodal lines.

Previously, we calculated some quantities of [M(dmdt)₂] (M = Pt, Ni) considering the spin–orbit interaction (SOI) as a parameter. There, we found that the SOI can reduce the Fermi surface in the bulk and induce helical edge modes.T.Kawamura2020 ; T.Kawamura2021 However, because the energy scale of SOI in this material ($\sim 0.0016$ eV) seems to be considerably smaller than the energy scale of the Fermi surface ($\sim 0.01$ eV),T.Kawamura2021 in reality SOI should not be large enough to significantly reduce the size of the Fermi surface. Therefore, we will omit the influence of SOI on the Knight shift and $1/T_{1}T$.

From the definition in Eq. 3, $H^{0}_{\alpha\beta,\sigma}(\textbf{k})$ are expressed by the following equations.

$$\begin{split}H^{0}_{AA,\sigma}(\textbf{k})&=2t_{9}\cos{k_{a}},\\ H^{0}_{AB,\sigma}(\textbf{k})&=t_{12}e^{i(-k_{a}+k_{b}+k_{c})}+t_{1}\\ &+t_{5}e^{i(k_{b}+k_{c})}+t_{4}e^{ik_{c}},\\ H^{0}_{AC,\sigma}(\textbf{k})&=t_{10}e^{i(-k_{a}+k_{b}+k_{c})}+t_{11}e^{i(k_{a}+k_{c})}+t_{6}\\ &+t_{3}e^{ik_{c}}+t_{2}e^{i(k_{b}+k_{c})},\\ H^{0}_{BB,\sigma}(\textbf{k})&=\Delta+2t_{7}\cos{(k_{b}+k_{c})}+2t_{8}\cos{k_{c}},\\ H^{0}_{BC,\sigma}(\textbf{k})&=t_{12}e^{i(-k_{a}+k_{b}+k_{c})}+t_{1}\\ &+t_{5}e^{i(k_{b}+k_{c})}+t_{4}e^{ik_{c}},\\ H^{0}_{CC,\sigma}(\textbf{k})&=2t_{9}\cos{k_{a}}.\end{split}$$

(4)

Here, for simplicity, we set all the lattice constants to be unity. Two of the three bands for which we performed Wannier fitting are occupied.T.Kawamura2020 Thus, the tight-binding model obtained using the Wannier fitting is also $2/3$ filling. The unperturbed Hamiltonian $\hat{H}^{0}_{\sigma}(\textbf{k})$ satisfies the eigenvalue equation

$$\hat{H}^{0}_{\sigma}(\textbf{k})\ket{{\textbf{k},n,\sigma}}=E_{n,\sigma}(\textbf{k})\ket{{\textbf{k},n,\sigma}},$$

(5)

$$\ket{{\textbf{k},n,\sigma}}=\left(\begin{array}[]{c}d_{A,n,\sigma}(\textbf{k})\\ d_{B,n,\sigma}(\textbf{k})\\ d_{C,n,\sigma}(\textbf{k})\end{array}\right).$$

(6)

where $E_{n,\sigma}(\textbf{k})$ is the eigenvalue and $\ket{{\textbf{k},n,\sigma}}$ is the eigen vector; $n$ is the band index; and $d_{\alpha,n,\sigma}(\textbf{k})$ denotes the wave functions. $\hat{H}^{0}_{\sigma}(\textbf{k})$ in Eq. 5 consists of the matrix elements $H^{0}_{\alpha\beta,\sigma}(\textbf{k})$ in Eqs. 2 and 4. Reflecting the $2/3$-filling, the chemical potential $\mu$ is determined by

$$\frac{1}{N_{L}}\sum_{\textbf{k},n,\sigma}f_{\textbf{k},n,\sigma}=4,$$

(7)

$$\epsilon_{\textbf{k},n,\sigma}\equiv E_{n,\sigma}(\textbf{k})-\mu,$$

(8)

$f_{\textbf{k},n,\sigma}$=$1/\left[1+\exp{(\epsilon_{\textbf{k},n,\sigma}/T)}\right]$ is the Fermi distribution function, and we have $\mu$=$E_{F}$ at $T$=$0$.

As to the second term in the Hamiltonian Eq. 2, we introduce the on-site repulsive interaction $U_{\alpha\alpha}$ defined on the fragment orbital $\alpha$, which is defined as the diagonal element of the interaction matrix as follows

	$\displaystyle\hat{U}$	$\displaystyle=$	$\displaystyle\left(\begin{array}[]{ccc}U_{AA}&0&0\\ 0&U_{BB}&0\\ 0&0&U_{CC}\\ \end{array}\right)$		(12)
		$\displaystyle=$	$\displaystyle\left(\begin{array}[]{ccc}\lambda U&0&0\\ 0&U&0\\ 0&0&\lambda U\\ \end{array}\right),$		(16)

Here, because of the inversion symmetry, we have $U_{AA}$=$U_{CC}$, and we use the relative size of $U_{AA}$ to $U_{BB}$, $\lambda$=$U_{AA}/U_{BB}$, as a control parameter of this model. According to respack, we found $U$=$6.72$ eV and $\lambda$=$0.79$ in the unscreened case, and $U$=$2.68$ eV and $\lambda$=$0.95$ in the screened case. In this study, we set $\lambda$=$0.95$ and use a value of $U$ less than $2.68$ eV because $\hat{U}$ tends to be overestimated in RPA.

The longitudinal and transverse spin susceptibilities are defined as follows:

	$\displaystyle\hat{\chi}^{zz}(\textbf{q},i\omega_{l})\equiv\frac{1}{2}\int^{1/T}_{0}d\tau e^{i\omega_{l}\tau}\left<T_{\tau}\hat{S}^{z}_{\textbf{q}}(\tau)\hat{S}^{z}_{-\textbf{q}}(0)\right>,$		(17)
	$\displaystyle\hat{S}^{z}_{\textbf{q}}=\frac{1}{N_{L}}\sum_{\textbf{k}}\left(\hat{c}^{\dagger}_{\textbf{k}+\textbf{q},\uparrow}\hat{c}_{\textbf{k},\uparrow}-\hat{c}^{\dagger}_{\textbf{k}+\textbf{q},\downarrow}\hat{c}_{\textbf{k},\downarrow}\right),$		(18)

	$\displaystyle\hat{\chi}^{\pm}(\textbf{q},i\omega_{l})\equiv\int^{1/T}_{0}d\tau e^{i\omega_{l}\tau}\left<T_{\tau}\hat{S}^{+}_{\textbf{q}}(\tau)\hat{S}^{-}_{-\textbf{q}}(0)\right>,$		(19)
	$\displaystyle\hat{S}^{+}_{\textbf{q}}=\frac{1}{N_{L}}\sum_{\textbf{k}}\hat{c}^{\dagger}_{\textbf{k}+\textbf{q},\uparrow}\hat{c}_{\textbf{k},\downarrow},$		(20)
	$\displaystyle\hat{S}^{-}_{-\textbf{q}}=\frac{1}{N_{L}}\sum_{\textbf{k}}\hat{c}^{\dagger}_{\textbf{k},\downarrow}\hat{c}_{\textbf{k}+\textbf{q},\uparrow}.$		(21)

Here, $i\omega_{l}$=$2li\pi T$ $(l\in\textbf{Z})$ is the Matsubara frequency and $\tau$ is the imaginary time. $\hat{S}^{z}_{\textbf{q}}$, $\hat{S}^{+}_{\textbf{q}}$, and $\hat{S}^{-}_{\textbf{q}}$ are the spin operators. $\hat{S}^{z}_{\textbf{q}}(\tau)$, $\hat{S}^{+}_{\textbf{q}}(\tau)$, and $\hat{S}^{-}_{\textbf{q}}(\tau)$ are described in the Heisenberg picture. The spin susceptibility is the proportionality coefficient of the magnetization to the infinitesimal magnetic field. It represents the degree of the “spin fluctuations”, because spins in the system sensitively respond to the infinitesimal magnetic field when spin susceptibility is large.
By performing a perturbation expansion of Eqs. 17 and 19, we obtain the non-interacting longitudinal spin susceptibility $\hat{\chi}^{zz,0}(\textbf{q},i\omega_{l})$ and non-interacting transverse spin susceptibility $\hat{\chi}^{\pm,0}(\textbf{q},i\omega_{l})$ as the zeroth-order perturbation terms. In this study, SU(2) symmetry is protected. Therefore, we define $\hat{\chi}^{zz,0}(\textbf{q},i\omega_{l})$=$\hat{\chi}^{\pm,0}(\textbf{q},i\omega_{l})$$\equiv$ $\hat{\chi}^{0}(\textbf{q},i\omega_{l})$. Its matrix elements are written as

$$\begin{split}&\chi^{0}_{\alpha\beta}(\textbf{q},i\omega_{l})\\ &=-\frac{T}{N_{L}}\sum_{\textbf{k},m}G^{0}_{\alpha\beta}(\textbf{k}+\textbf{q},i\tilde{\omega}_{m}+i\omega_{l})G^{0}_{\beta\alpha}(\textbf{k},i\tilde{\omega}_{m}),\end{split}$$

(22)

$$\begin{split}G^{0}_{\alpha\beta,\sigma}(\textbf{k},i\tilde{\omega}_{l})=\sum_{n}d_{\alpha,n,\sigma}(\textbf{k})d^{*}_{\beta,n,\sigma}(\textbf{k})\frac{1}{i\tilde{\omega}_{l}-\epsilon_{\textbf{k},n,\sigma}}.\end{split}$$

(23)

Eq. 23 expresses the matrix elements of the non-interacting Green’s function. $i\tilde{\omega}_{l}$=$(2l+1)i\pi T$ is the Matsubara frequency of the fermions. The spin index $\sigma$ is omitted in Eq. 22 because we have $\epsilon_{\textbf{k},n,\uparrow}$=$\epsilon_{\textbf{k},n,\downarrow}$ in Eq. 23. The longitudinal and transverse spin susceptibilities in RPA are represented by the Feynman diagrams shown in Fig. 4.

The first terms in the right-hand sides of diagrams (A) and (B) correspond to the terms $\hat{\chi}^{zz,0}(\textbf{q},i\omega_{l})$=$\hat{\chi}^{\pm,0}(\textbf{q},i\omega_{l})$$=$ $\hat{\chi}^{0}(\textbf{q},i\omega_{l})$ (Eq. 22). Because the interacting longitudinal and transverse spin susceptibilities are represented by summations of the series of $\hat{U}\hat{\chi}^{0}(\textbf{q},i\omega_{l})$ in RPA, they are written as

	$\displaystyle\hat{\chi}^{zz,s}(\textbf{q},i\omega_{l})$	$\displaystyle=$	$\displaystyle\hat{\chi}^{\pm,s}(\textbf{q},i\omega_{l})\equiv\hat{\chi}^{s}(\textbf{q},i\omega_{l})$
		$\displaystyle=$	$\displaystyle\hat{\chi}^{0}(\textbf{q},i\omega_{l})[\hat{I}-\hat{U}\hat{\chi}^{0}(\textbf{q},i\omega_{l})]^{-1}$
	,

where $\hat{I}$ is the unit matrix.

Here we introduce the Stoner factor $\xi_{s}(\textbf{q})$ representing the degree of enhancement of the spin fluctuations. The Stoner factor $\xi_{s}(\textbf{q})$ is defined as the maximum eigenvalue of $\hat{U}\hat{\chi}^{0}(\textbf{q},0)$. The relation between $\xi_{s}(\textbf{q})$ and $\hat{\chi}^{s}(\textbf{q},0)$ in the three-orbital model is given by

$\displaystyle\hat{\chi}^{s}(\textbf{q},0)=\frac{1}{(1-\xi_{s}(\textbf{q}))}\frac{\hat{\chi}^{0}(\textbf{q},0)\hat{P}(\textbf{q})}{(1-\phi_{1}(\textbf{q}))(1-\phi_{2}(\textbf{q}))},$

(25)

where $\xi_{s}(\textbf{q})$, $\phi_{1}(\textbf{q})$, and $\phi_{2}(\textbf{q})$ are the maximum and other eigenvalues of $\hat{U}\hat{\chi}^{0}(\textbf{q},0)$. $\hat{P}(\textbf{q})$ is the adjugate matrix of $\hat{I}-\hat{U}\hat{\chi}^{0}(\textbf{q},0)$. The eigenvalues of $\hat{U}\hat{\chi}^{0}(\textbf{q},0)$ is smaller than $1$ in the paramagnetic regime. When $\xi_{s}(\textbf{q})\rightarrow 1$, the spin susceptibility $\hat{\chi}^{s}(\textbf{q},0)$ diverges and induces a magnetic order, corresponding to the wavenumber q.

Within the framework of linear response theory, the Knight shift, $K$, and the spin-lattice relaxation rate, $(1/T_{1}T)$, for the orbital $\alpha$ are given byS.Katayama2009

$\displaystyle K_{\alpha}$

$\displaystyle\propto$

$\displaystyle\sum_{\beta}{\rm Re}\left[\chi^{zz}_{\alpha\beta}(\textbf{0},0)\right],$

(26)

$\displaystyle\left(1/T_{1}T\right)_{\alpha}$

$\displaystyle\propto$

$\displaystyle\lim_{\omega\rightarrow+0}\left[\frac{1}{N_{L}}\sum_{\textbf{q}}\frac{{\rm Im}\chi^{\pm}_{\alpha\alpha}(\textbf{q},\omega)}{\omega}\right]$

(27)

Here, note that Eq. II is satisfied. According to Eqs. 25 and 27, all the q components for which $\xi_{s}(\textbf{q})$ becomes close to unity make a dominant contribution to $1/T_{1}T$ because they lead to a large value in the spin susceptibility. By contrast, the Knight shift is solely affected by the q=0 component that satisfies $\xi_{s}(\textbf{q})\sim 1$ (see Eq. 26).

Because the spin susceptibility in the real-frequency representation $\hat{\chi}^{s}(\textbf{q},\omega)$ is necessary for $1/T_{1}T$, we obtain $\hat{\chi}^{s}(\textbf{q},\omega)$ by performing an analytic continuation of Eq. II. In this way, we use the real-frequency representation depending on the physical quantities.

In this section, we calculate the electronic state, spin susceptibility, Knight shift, and $1/T_{1}T$ in the absence of the repulsive interaction $U$. Further, we find that the spin susceptibility in [Ni(dmdt)₂] greatly depends on the fragment orbitals and will explain the relationship between the spin susceptibilities and the wave functions.

In this section, we calculate spin fluctuations, the Knight shift, and $1/T_{1}T$ in the presence of $U$. It is shown that electron correlation effects are important for explaining the observed temperature dependence of the Knight shift and $1/T_{1}T$ reported by ¹³C-NMR experiments.T.Sekine.Private According to Eq. 25, the Stoner factor $\xi_{s}(\textbf{q})$ that has a value close to unity gives major contributions to the spin susceptibility and therefore to the Knight shift and $1/T_{1}T$. $\xi_{s}(\textbf{q}=\textbf{0})\approx 1$ contributes to the Knight shift (Eq. 26) and $1/T_{1}T$. $\xi_{s}(\textbf{q}=\textbf{Q})\approx 1$ does not contribute to the Knight shift but to $1/T_{1}T$ (Eq. 27).