Effective theory of monolayer TMDC double quantum dots

We give here the definitions for the different terms of the Hamiltonians that describe the various scenarios studied in this work. The operators $c_{j\tau\sigma}^{(\dagger)}$ annihilate (create) an electron in QD j with valley τ and spin σ. Here j  =  L (R) refers to the left (right) QD, $\newcommand{\ku}{K} \tau = \ku$ ($\newcommand{\kd}{\overline{K}} \kd$) indicates the positive (negative) valley and $\newcommand{\su}{\uparrow} \sigma = \su$ ($\newcommand{\sd}{\downarrow} \sd$) specifies spin up (spin down).

The on-site Coulomb repulsion between electrons in the same QD is captured by the Hubbard Hamiltonian,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\hamhubb}{H_U} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\su}{\uparrow} \displaystyle \label{eqn:hamhubb} \hamhubb = \frac{U}{2} \sum_{j = L,R} n_j (n_j - 1), \nonumber \end{align} \tag{ 1 }$$

where U  >  0 is the positive charging energy of the dot and the number operator is defined as

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\su}{\uparrow} \newcommand{\ann}[1]{c_{#1}^{}} \newcommand{\cre}[1]{c_{#1}^\dagger} \displaystyle n_j = \sum_{\tau = K, \overline{K}} \sum_{\sigma = \uparrow, \downarrow} \cre{j\tau\sigma} \ann{j\tau\sigma}. \nonumber \end{align} \tag{ 2 }$$

A detuning term specifies the energy difference ε between the dots,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\hamdetun}{H_\varepsilon} \newcommand{\h}[2]{h_{#1}^{#2}} \displaystyle \label{eqn:hamdetun} \hamdetun = \frac{\varepsilon}{2} (n_L - n_R). \nonumber \end{align} \tag{ 3 }$$

Electron-hopping from one dot to the other is accounted for by a tunneling term that preserves spin and valley,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\hamtunn}{H_t} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\su}{\uparrow} \newcommand{\ann}[1]{c_{#1}^{}} \newcommand{\cre}[1]{c_{#1}^\dagger} \displaystyle \label{eqn:hamtunn} \hamtunn = \sum_{\tau, \sigma} \left (t \, \cre{R\sigma\tau} \ann{L\sigma\tau} + {\rm h.c.} \right), \nonumber \end{align} \tag{ 4 }$$

where the tunneling coefficient t is generally a complex number.

The intrinsic spin–orbit coupling is modeled by a simple time-reversal symmetric ($\mathcal{T}$-symmetric) spin-splitting: $\Delta \tau_z \sigma_z$ [7]. Here, $\tau_i$ ($\sigma_i$) is the ith Pauli matrix acting on the valley (spin) DOF ($i = x, y, z$), while Δ is a real, positive or negative, coupling constant. This implies that the Kramers pair states in the set $\newcommand{\ku}{K} \newcommand{\kd}{\overline{K}} \newcommand{\su}{\uparrow} \newcommand{\sd}{\downarrow} \renewcommand{\ket}[1]{|#1\rangle} \mathcal{P} = \left \{\ket{\ku\su}, \ket{\kd\sd} \right \}$ are shifted by the energy $+\Delta$, while the Kramers pair states in the set $\newcommand{\ku}{K} \newcommand{\kd}{\overline{K}} \newcommand{\su}{\uparrow} \newcommand{\sd}{\downarrow} \renewcommand{\ket}[1]{|#1\rangle} \mathcal{N} = \left \{\ket{\ku\sd}, \ket{\kd\su} \right \}$ are shifted by the energy $-\Delta$. We call $\mathcal{P}$ the positive Kramers pair and $\mathcal{N}$ the negative Kramers pair. For the double dot system,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\hamspinsplit}{H_\Delta} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\su}{\uparrow} \newcommand{\ann}[1]{c_{#1}^{}} \newcommand{\cre}[1]{c_{#1}^\dagger} \displaystyle \label{eqn:spinsplitting} \hamspinsplit = \Delta \sum_{j,\tau,\sigma} \cre{j\tau\sigma} (\tau_z)_{\tau\tau} (\sigma_z)_{\sigma\sigma} \ann{j\tau\sigma}. \nonumber \end{align} \tag{ 5 }$$

Equation (5) assumes that the spin–orbit splitting is the same for every dot, which is usually the case for dots created on the same material. In case the spin–orbit splitting is different we use the following generalisation,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\hamspinsplitLR}{H_{\Delta_L,\Delta_R}} \newcommand{\hamspinsplit}{H_\Delta} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\su}{\uparrow} \newcommand{\ann}[1]{c_{#1}^{}} \newcommand{\cre}[1]{c_{#1}^\dagger} \displaystyle \label{eqn:spinsplittingLR} \hamspinsplitLR = \sum_j \Delta_j \sum_{\tau,\sigma} \cre{j\tau\sigma} (\tau_z)_{\tau\tau} (\sigma_z)_{\sigma\sigma} \ann{j\tau\sigma}, \nonumber \end{align} \tag{ 6 }$$

where $\Delta_L$ and $\Delta_R$ are the spin–orbit splittings in the left and right QD respectively.

We also consider the coupling of spin and valley to an external magnetic field. The corresponding spin Zeeman term is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\hamspinmix}{H_S} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\vec{t}}[1]{\boldsymbol{#1}} \newcommand{\su}{\uparrow} \newcommand{\ann}[1]{c_{#1}^{}} \newcommand{\cre}[1]{c_{#1}^\dagger} \displaystyle \label{eqn:spinmixing} \hamspinmix = \sum_j \vec{t}{h}_{Sj} \cdot \sum_{\tau,\sigma_1,\sigma_2} \cre{j\tau\sigma_1} (\vec{t}{\sigma})_{\sigma_1\sigma_2} \ann{j\tau\sigma_2}, \nonumber \end{align} \tag{ 7 }$$

where $\newcommand{\vec{t}}[1]{\boldsymbol{#1}} \vec{t}{h}_{SL}$ and $\newcommand{\vec{t}}[1]{\boldsymbol{#1}} \vec{t}{h}_{SR}$ are two vectors of coupling constants for left and right QD respectively and $\newcommand{\vec{t}}[1]{\boldsymbol{#1}} \vec{t}{\sigma}$ is the vector of Pauli matrices acting on spin.

The valley Zeeman term is the valley counterpart of the spin Zeeman but considering only the z-Pauli matrix acting on the valley. This is motivated by [7] and recent experiments [47, 48],

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\hamvalleyzeeman}{H_V} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\su}{\uparrow} \newcommand{\ann}[1]{c_{#1}^{}} \newcommand{\cre}[1]{c_{#1}^\dagger} \displaystyle \label{eqn:valleyzeeman} \hamvalleyzeeman = \sum_j h_{Vjz} \sum_{\tau,\sigma} \cre{j\tau\sigma} (\tau_z)_{\tau\tau} \ann{j\tau\sigma} \nonumber \end{align} \tag{ 8 }$$

where $h_{VLz}$ and $h_{VRz}$ describe the valley splittings in the left and right dot. A concise outline of the interactions and the DOF of our DQD model is shown in figure 1.

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We consider the case where there are two electrons in the system. The possible charge configurations are $(2, 0)$, $(1, 1)$ and $(0, 2)$, where $(n_L, n_R)$ means that there are n_L electrons in the left QD and n_R electrons in the right QD. Because of the spin and valley degrees of freedom and due to the Pauli exclusion principle, there are 28 linearly independent states: 6 $(2, 0)$-states, 16 $(1, 1)$-states and 6 $(0, 2)$-states. Throughout this paper, we always assume a small detuning and weak tunneling, i.e.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eqn:mainswcond} |t| \ll |U \pm \varepsilon|. \nonumber \end{align} \tag{ 9 }$$

It is natural to refer to certain operators that will appear as projectors on specific states. We present a convention to name all 28 states of the total Hilbert space and we give the projection operators that will be useful later.

Because of the Pauli exclusion principle, not all the states in the $(1, 1)$-subspace are allowed to tunnel to $(0, 2)$ or $(2, 0)$-states, but only those which are antisymmetric in spin and valley [44, 49]. Following the antisymmetric nature of the tunneling states, we begin by introducing a basis consisting of states that are symmetric or antisymmetric in both spin and valley: $\renewcommand{\ket}[1]{|#1\rangle} \ket{s_V s_S}_{(n_L, n_R)}$, where $(n_L, n_R)$ is the charge configuration and $s_V$ (s_S) indicates the exchange symmetry of valley (spin) DOF. The exchange symmetry can be either that of a singlet (S) or that of a triplet (T₋, T₀, T₊); see table 1 for the definitions of these states. When $(n_L, n_R) = (1, 1)$ we omit the indication of the charge configuration in the subscript. The equal spin–orbit coupling term $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \hamspinsplit$ (equation (5)) is not diagonal in this basis. In order to work with a basis that makes $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \hamspinsplit$ diagonal, in the $(1, 1)$-subspace we substitute $\renewcommand{\ket}[1]{|#1\rangle} \ket{S T_0}$, $\renewcommand{\ket}[1]{|#1\rangle} \ket{T_0 S}$, $\renewcommand{\ket}[1]{|#1\rangle} \ket{S S}$ and $\renewcommand{\ket}[1]{|#1\rangle} \ket{T_0 T_0}$ with the following states (see appendix A),

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\nup}{n} \displaystyle \ket{\nup_\pm} = (\ket{S T_0} \pm \ket{T_0 S})/\sqrt{2}, \nonumber \end{align} \tag{ 10a }$$

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\nd}{\overline{n}} \displaystyle \ket{\nd_\pm} = (\ket{T_0 T_0} \pm \ket{S S})/\sqrt{2}. \nonumber \end{align} \tag{ 10b }$$

Table 1. Definitions of singlet and triplet states for the spin and valley DOF.

	$\renewcommand{\ket}[1]{|#1\rangle} \ket{S}$	$\renewcommand{\ket}[1]{|#1\rangle} \ket{T_-}$	$\renewcommand{\ket}[1]{|#1\rangle} \ket{T_0}$	$\renewcommand{\ket}[1]{|#1\rangle} \ket{T_+}$
Spin	$\newcommand{\su}{\uparrow} \newcommand{\sd}{\downarrow} \renewcommand{\ket}[1]{|#1\rangle} \frac{\ket{\su\sd}-\ket{\sd\su}}{\sqrt{2}}$	$\newcommand{\sd}{\downarrow} \renewcommand{\ket}[1]{|#1\rangle} \ket{\sd\sd}$	$\newcommand{\su}{\uparrow} \newcommand{\sd}{\downarrow} \renewcommand{\ket}[1]{|#1\rangle} \frac{\ket{\su\sd}+\ket{\sd\su}}{\sqrt{2}}$	$\vert\uparrow\uparrow\rangle$
Valley	$\newcommand{\ku}{K} \newcommand{\kd}{\overline{K}} \renewcommand{\ket}[1]{|#1\rangle} \frac{\ket{\ku\kd}-\ket{\kd\ku}}{\sqrt{2}}$	$\newcommand{\kd}{\overline{K}} \renewcommand{\ket}[1]{|#1\rangle} \ket{\kd\kd}$	$\newcommand{\ku}{K} \newcommand{\kd}{\overline{K}} \renewcommand{\ket}[1]{|#1\rangle} \frac{\ket{\ku\kd}+\ket{\kd\ku}}{\sqrt{2}}$	$\newcommand{\ku}{K} \renewcommand{\ket}[1]{|#1\rangle} \ket{\ku\ku}$

Here $\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nup_\pm}$ ($\newcommand{\nd}{\overline{n}} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nd_\pm}$) are antisymmetric (symmetric) spin-valley states which are superpositions of only positive (subscript  +) or negative (subscript  −) Kramers pairs, see also appendix B. Note that $\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nup_\pm}$ ($\newcommand{\nd}{\overline{n}} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nd_\pm}$) are odd (even) under the time-reversal operator $\mathcal{T}$. For these states there is no defined exchange symmetry for spin or valley alone. States analogous to $\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nup_\pm}$ are defined in the $(2, 0)$ and $(0, 2)$-subspaces. In table 2, we list all these states grouped by charge configuration and by symmetry under exchange.

Table 2. The first column on the left hand side reports the value of Coulomb repulsion (equation (1)) and detuning (equation (3)) for the different charge configurations. The three columns on the right hand side show which states are shifted by $-2\Delta$, 0 and $+2\Delta$ by the action of the symmetric spin–orbit coupling $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \hamspinsplit$ defined in equation (5). The states are grouped as $(1, 1)$-states (antisymmetric and symmetric), $(2, 0)$-states and $(0, 2)$-states (only antisymmetric).

$\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamhubb}{H_U} \newcommand{\hamdetun}{H_\varepsilon} \hamhubb + \hamdetun$	$-2\Delta$	0	$+2\Delta$
$(1, 1)$-subspace, antisymmetric
0	$\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nup_-}$	$\newcommand{\fu}{T_{+}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fu\au}$	$\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nup_+}$
		$\newcommand{\fd}{T_{-}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fd\au}$
		$\newcommand{\fu}{T_{+}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\au\fu}$
		$\newcommand{\fd}{T_{-}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\au\fd}$
$(1, 1)$-subspace, symmetric
0	$\newcommand{\nd}{\overline{n}} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nd_-}$	$\newcommand{\fu}{T_{+}} \newcommand{\adddd}{T_0} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fu\adddd}$	$\newcommand{\nd}{\overline{n}} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nd_+}$
	$\newcommand{\fu}{T_{+}} \newcommand{\fd}{T_{-}} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fu\fd}$	$\newcommand{\fd}{T_{-}} \newcommand{\adddd}{T_0} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fd\adddd}$	$\newcommand{\fu}{T_{+}} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fu\fu}$
	$\newcommand{\fu}{T_{+}} \newcommand{\fd}{T_{-}} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fd\fu}$	$\newcommand{\fu}{T_{+}} \newcommand{\adddd}{T_0} \renewcommand{\ket}[1]{|#1\rangle} \ket{\adddd\fu}$	$\newcommand{\fd}{T_{-}} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fd\fd}$
		$\newcommand{\fd}{T_{-}} \newcommand{\adddd}{T_0} \renewcommand{\ket}[1]{|#1\rangle} \ket{\adddd\fd}$
$(2, 0)$-subspace
$U+\varepsilon$	$\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nup_-}_{(2, 0)}$	$\newcommand{\fu}{T_{+}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fu\au}_{(2, 0)}$	$\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nup_+}_{(2, 0)}$
		$\newcommand{\fd}{T_{-}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fd\au}_{(2, 0)}$
		$\newcommand{\fu}{T_{+}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\au\fu}_{(2, 0)}$
		$\newcommand{\fd}{T_{-}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\au\fd}_{(2, 0)}$
$(0, 2)$-subspace
$U-\varepsilon$	$\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nup_-}_{(0, 2)}$	$\newcommand{\fu}{T_{+}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fu\au}_{(0, 2)}$	$\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nup_+}_{(0, 2)}$
		$\newcommand{\fd}{T_{-}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fd\au}_{(0, 2)}$
		$\newcommand{\fu}{T_{+}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\au\fu}_{(0, 2)}$
		$\newcommand{\fd}{T_{-}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\au\fd}_{(0, 2)}$

Projection operators on each one of the antisymmetric $(1, 1)$-states are easy to obtain (see appendix B),

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\vec{t}}[1]{\boldsymbol{#1}} \newcommand{\id}{\mathbb{1}} \newcommand{\au}{S} \newcommand{\fufd}{T_{\pm}} \newcommand{\fu}{T_{+}} \displaystyle \label{eqn:projfufdau} P_{\ket{\fufd\au}} = \frac{1}{16} \left (\id \pm \tau_{Lz} \right) \left (\id \pm \tau_{Rz} \right) \left (\id - \vec{t}{\sigma}_{L} \cdot \vec{t}{\sigma}_{R} \right), \nonumber \end{align} \tag{ 11a }$$

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\vec{t}}[1]{\boldsymbol{#1}} \newcommand{\id}{\mathbb{1}} \newcommand{\au}{S} \newcommand{\fufd}{T_{\pm}} \newcommand{\fu}{T_{+}} \displaystyle \label{eqn:projaufufd} P_{\ket{\au\fufd}} = \frac{1}{16} \left (\id \pm \sigma_{Lz} \right) \left (\id \pm \sigma_{Rz} \right) \left (\id - \vec{t}{\tau}_{L} \cdot \vec{t}{\tau}_{R} \right), \nonumber \end{align} \tag{ 11b }$$

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\id}{\mathbb{1}} \newcommand{\nup}{n} \displaystyle \label{eqn:projnup} P_{\ket{\nup_\pm}} & = \frac{1}{16} \left (\id - \tau_{Lz} \tau_{Rz} \right) \left (\id - \sigma_{Lz} \sigma_{Rz} \right) \nonumber \\ &\quad \times \left (\id \pm \tau_{Lz} \sigma_{Lz} \right) \left (\id - \tau_{Lx} \sigma_{Lx} \tau_{Rx} \sigma_{Rx} \right), \nonumber \end{align} \tag{ 11c }$$

where $\tau_{ji}$ ($\sigma_{ji}$) is the ith Pauli matrix acting on valley (spin) in QD $j = L, R$ and $\newcommand{\vec{t}}[1]{\boldsymbol{#1}} \vec{t}{\sigma}_j$, $\newcommand{\vec{t}}[1]{\boldsymbol{#1}} \vec{t}{\tau}_j$ are vectors of Pauli matrices acting on spin or valley respectively. We note that the projection operator over the whole antisymmetric subspace of the $(1, 1)$-sector can be written as [45] (see appendix B),

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\vec{t}}[1]{\boldsymbol{#1}} \displaystyle \label{eqn:antisymmproj} P_{\rm{as}} = (3 - \vec{t}{\sigma}_L \cdot \vec{t}{\sigma}_R - \vec{t}{\tau}_L \cdot \vec{t}{\tau}_R - (\vec{t}{\sigma}_L \cdot \vec{t}{\sigma}_R)(\vec{t}{\tau}_L \cdot \vec{t}{\tau}_R)) / 8. \nonumber \end{align} \tag{ 12 }$$

We finish this section by briefly considering the case where there is no spin–orbit splitting ($\Delta_L = \Delta_R = 0$). The total Hamiltonian is, then,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\hamtunn}{H_t} \newcommand{\hamdetun}{H_\varepsilon} \newcommand{\hamhubb}{H_U} \newcommand{\hamtot}{H_{\rm{tot}}} \newcommand{\h}[2]{h_{#1}^{#2}} \displaystyle \hamtot = \hamhubb + \hamdetun + \hamtunn, \nonumber \end{align} \tag{ 13 }$$

where $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamhubb}{H_U} \hamhubb$, $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamdetun}{H_\varepsilon} \hamdetun$ and $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamtunn}{H_t} \hamtunn$ are defined in equations (1), (3) and (4) respectively. This model describes DQD systems with fourfold degenerate spin and valley states in each dot (and no spin–orbit interaction). This case has been extensively treated in [45]. For later reference and readability we report the resulting effective Hamiltonian,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eqn:heffexchangeonly} H_{\rm{ef\,f}} = -J P_{\rm{as}}, \nonumber \end{align} \tag{ 14 }$$

where the exchange energy J in this case has the usual form,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eqn:exchangeenergy} J = \frac{4 |t|^2 U}{U^2 - \varepsilon^2}, \nonumber \end{align} \tag{ 15 }$$

and the definition of $P_{\rm{as}}$ is given in equation (12). This Hamiltonian shifts down the energy of all the (1,1) states in the antisymmetric subspace by  −J. It means that the ground state is any state in this 6-dimensional subspace and it has energy  −J. The first excited states are those in the orthogonal 10-dimensional symmetric subspace, with energy 0. This is represented in figure 2(a).

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We first consider the case where the spin–orbit coupling is equal for both QDs: $\Delta_L = \Delta_R = \Delta$. We call this symmetric spin–orbit splitting. The total Hamiltonian is

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\hamspinsplit}{H_\Delta} \newcommand{\hamtunn}{H_t} \newcommand{\hamdetun}{H_\varepsilon} \newcommand{\hamhubb}{H_U} \newcommand{\hamtot}{H_{\rm{tot}}} \newcommand{\h}[2]{h_{#1}^{#2}} \displaystyle \label{eqn:symmspinsplit} \hamtot = \hamhubb + \hamdetun + \hamtunn + \hamspinsplit, \nonumber \end{align} \tag{ 16 }$$

where $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \hamspinsplit$ is defined in equation (5). This Hamiltonian describes DQDs created in TMDCs [3–7, 9], but it may also be used to study DQDs in CNTs [25, 42–44, 50, 51].

Symmetric spin–orbit coupling $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \hamspinsplit$ shifts those states formed by two elements of the negative (positive) Kramers pair by $-2\Delta$ ($+2\Delta$), while it leaves unchanged those states formed by one element of the negative Kramers pair and one element of the positive Kramers pair (see table 2). In order to identify the $(1, 1)$-sector as our low energy subspace (LES) we have to guarantee first of all that no $(2, 0)$ or $(0, 2)$-state is lower in energy than any $(1, 1)$-state. Looking at table 2 we see that this condition is met when

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eqn:symmspinsplitcond} 4 |\Delta| < U-|\varepsilon|. \nonumber \end{align} \tag{ 17 }$$

When both (9) and (17) are satisfied, it follows that $(2, 0)$ and $(0, 2)$-states are energetically unfavored and tunneling out from the $(1, 1)$-sector is strongly suppressed. However, virtual tunneling processes must be taken into account. It is important to notice that antisymmetric $(1, 1)$-states can tunnel only to their $(0, 2)$ and $(2, 0)$ counterpart states that have the same spin and valley configuration. This is due to the spin- and valley-preserving nature of the tunneling term of equation (4): there is no transition from a negative Kramers pair to a positive Kramers pair and viceversa. Therefore, all the energy differences between initial and final states in a virtual tunneling process do not depend on Δ and the exchange interaction that emerges does not change from the case of zero spin–orbit splitting reported in section 2.3. Exchange interaction and symmetric spin–orbit coupling act independently of each other. We obtain the effective Hamiltonian:

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\hameff}{H_{\rm{ef\,f}}} \newcommand{\h}[2]{h_{#1}^{#2}} \displaystyle \label{eqn:heffsymmetric} \hameff = -JP_{\rm{as}} + \Delta \Sigma, \nonumber \end{align} \tag{ 18 }$$

where J is the same constant defined in (15), $P_{\rm{as}}$ is given in (12), $\newcommand{\where}[2]{\left . {#1} \right |_{#2}} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \Delta \Sigma = \where{\hamspinsplit}{(1, 1)} = \Delta \left (\tau_{Lz} \sigma_{Lz} + \tau_{Rz} \sigma_{Rz} \right)$ and $\newcommand{\where}[2]{\left . {#1} \right |_{#2}} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \where{\hamspinsplit}{(1, 1)}$ is the restriction of $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \hamspinsplit$ to the $(1, 1)$-subspace.

It is simple to evaluate the consequences of equation (18) since it is diagonal in the states of table 2. Assuming now that $\Delta > 0$, the ground state is $\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nup_-}$ because it is the only state that is shifted down by both  −J and $-2\Delta$, thus the ground state space is one dimensional, see figures 2(b) and (c). The dimensionality of the ground state space changes from one to six only when Δ is exactly equal to zero (see figure 2(a)) and this degeneracy is lifted linearly in Δ. As shown in figures 2(b) and (c), there are five groups of excited states with various degrees of degeneracy and whose relative distances in energy depend on the sizes of Δ and J. For later comparison, we observe that in this case each antisymmetric state is separated from its symmetric counterpart by the same energy J. In the case $\Delta < 0$, the results are analogous, but the ground state is $\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nup_+}$.

Regarding the experimental situation, the few available experiments indicate $U \approx 2$ meV for QDs created in WSe₂ [37] and MoS₂ [39–41] with QD radii around 100 nm. On the other hand, the theory predicts $2\Delta$ to be about 3 meV for MoS₂ and larger for other compounds [7, 52]. Therefore, at the moment, condition (17) seems not to be satisfied for TMDCs, but further experiments with smaller QD sizes may lead to higher charging energies. However, when equation (17) is not satisfied, a consistent effective theory can be derived in a subspace of the $(1, 1)$-sector, as explained in section 4.

Suppose now that the spin–orbit strength is different in the two dots, $\Delta_L \neq \Delta_R$, this is the case of asymmetric spin–orbit splitting. The total Hamiltonian is

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\hamspinsplitLR}{H_{\Delta_L,\Delta_R}} \newcommand{\hamspinsplit}{H_\Delta} \newcommand{\hamtunn}{H_t} \newcommand{\hamdetun}{H_\varepsilon} \newcommand{\hamhubb}{H_U} \newcommand{\hamtot}{H_{\rm{tot}}} \newcommand{\h}[2]{h_{#1}^{#2}} \displaystyle \label{eqn:asymmspinsplit} \hamtot = \hamhubb + \hamdetun + \hamtunn + \hamspinsplitLR, \nonumber \end{align} \tag{ 19 }$$

where $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \hamspinsplit$ has been substituted by $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \newcommand{\hamspinsplitLR}{H_{\Delta_L,\Delta_R}} \hamspinsplitLR$ defined in equation (6).

There is theoretical evidence that for (rather small) QDs on TMDCs the spin–orbit strength Δ depends on the radius of the QD [33]. Therefore, the Hamiltonian in equation (19) can reflect a situation where the two dots have different sizes. Changing the size of the QD would allow for a smooth and tunable adjustment of Δ. Another situation where $\Delta_L \neq \Delta_R$ can be relevant is when the dots would be created in lateral heterojunctions such that each dot is created in different types of TMDC, which intrinsically possess different spin–orbit splittings [52]. Lateral heterojunctions of different TMDCs have already been demonstrated by several groups [53–55]. To our knowledge no experiment has been reported yet involving such a DQD. We notice that for heterojunctions the situation might be further complicated by mismatch in the energy band gap, acting as a detuning away from the interface, and by likely non-negligible differences in the Coulomb charging energy U of the two compounds.

We point out that $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \newcommand{\hamspinsplitLR}{H_{\Delta_L,\Delta_R}} \hamspinsplitLR$ is not diagonal in the basis of the states listed in table 2. Each of the antisymmetric states $\newcommand{\fu}{T_{+}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fu\au}$, $\newcommand{\fd}{T_{-}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fd\au}$, $\newcommand{\fu}{T_{+}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\au\fu}$, $\newcommand{\fd}{T_{-}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\au\fd}$ is mixed with its symmetric counterpart and the off-diagonal elements are $\pm (\Delta_L - \Delta_R)$. This is relevant to determine the ground state, but not for the form of the effective Hamiltonian, because the mixing happens inside the $(1, 1)$-subspace. Another difference is that, in contrast to table 2, the $\pm 2\Delta$ on the diagonal of the Hamiltonian are replaced by $\pm (\Delta_L + \Delta_R)$ in the $(1, 1)$-subspace, by $\pm 2\Delta_L$ in $(2, 0)$ and by $\pm 2\Delta_R$ in $(0, 2)$. When condition (9) is also valid, we can ensure that $(1, 1)$-states are lower in energy than $(2, 0)$ and $(0, 2)$ states by requiring that

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle & \max\{|\Delta_L + \Delta_R|, |\Delta_L - \Delta_R|\} \nonumber \\ & + 2 \max\{|\Delta_L|, |\Delta_R|\} < U - |\varepsilon|. \nonumber \end{align} \tag{ 20 }$$

In this case we can use the Schrieffer–Wolff transformation [56–58] to obtain the effective Hamiltonian. The term $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \newcommand{\hamspinsplitLR}{H_{\Delta_L,\Delta_R}} \hamspinsplitLR$ has three significant consequences. Firstly, the following matrix element differences now depend on $\pm (\Delta_L - \Delta_R)$,

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\nup}{n} \displaystyle E_{\ket{\nup_\pm}} - E_{\ket{\nup_\pm}_{(0,2)}} = \pm (\Delta_L - \Delta_R) - U + \varepsilon, \nonumber \end{align} \tag{ 21a }$$

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\nup}{n} \displaystyle E_{\ket{\nup_\pm}} - E_{\ket{\nup_\pm}_{(2,0)}} = \mp (\Delta_L - \Delta_R) - U - \varepsilon, \nonumber \end{align} \tag{ 21b }$$

where $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamtot}{H_{\rm{tot}}} \renewcommand{\bra}[1]{\langle#1|} \renewcommand{\ket}[1]{|#1\rangle} E_{\ket{\psi}} = \bra{\psi}\hamtot\ket{\psi}$. Secondly, in order to apply perturbation theory to the $(1, 1)$-subspace, the coupling constants must satisfy the conditions $|t| \ll | U \pm \varepsilon |$, $|t| \ll | U \pm (\varepsilon + \Delta_L - \Delta_R) |$ and $|t| \ll | U \pm (\varepsilon - \Delta_L + \Delta_R) |$ simultaneously. In turn, these imply that $| \Delta_L - \Delta_R | \ll U$. Lastly, the exchange energies relative to $\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nup_\pm}$ acquire a dependence on the asymmetry of the spin–orbit splittings,

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\nup}{n} \displaystyle \label{eqn:asymmexchangeenergy} J_{\ket{\nup_\pm}} = \frac{4 |t|^2 U}{(U + \varepsilon \pm \Delta_L \mp \Delta_R) (U - \varepsilon \mp \Delta_L \pm \Delta_R)}, \nonumber \end{align} \tag{ 22 }$$

see equation (15). The exchange energies for states $\newcommand{\fu}{T_{+}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fu\au}$, $\newcommand{\fd}{T_{-}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fd\au}$, $\newcommand{\fu}{T_{+}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\au\fu}$ and $\newcommand{\fd}{T_{-}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\au\fd}$ are not affected.

An effective Hamiltonian can be written in a compact form in the following way,

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\hameff}{H_{\rm{ef\,f}}} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\nup}{n} \displaystyle \label{eqn:heffasymm} \hameff =&\, -J P_{\rm{as}} + \Delta_L \Sigma_L + \Delta_R \Sigma_R- (J_{\ket{\nup_-}} - J) P_{\ket{\nup_-}} \nonumber \\ & - (J_{\ket{\nup_+}} - J) P_{\ket{\nup_+}}, \nonumber \end{align} \tag{ 23 }$$

where $\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} P_{\ket{\nup_-}}$ and $\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} P_{\ket{\nup_+}}$ are the projectors on $\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nup_-}$ and $\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nup_+}$ defined in equation (11c) and in this case we write the restriction $\newcommand{\where}[2]{\left . {#1} \right |_{#2}} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \newcommand{\hamspinsplitLR}{H_{\Delta_L,\Delta_R}} \where{\hamspinsplitLR}{(1, 1)} = \Delta_L \Sigma_L + \Delta_R \Sigma_R,$ with $\Sigma_j = \tau_{jz} \sigma_{jz}$. By diagonalizing the Hamiltonian (23), we obtain the energy levels shown in figure 3. Interestingly, the two levels that appear at energy 0 and energy  −J for the symmetric spin–orbit splitting case move to the energies,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eqn:offdiageigenval} -\frac{J}{2} \pm \frac{1}{2} \sqrt{J^2 + 4(\Delta_L - \Delta_R)^2} = -\frac{J}{2} \pm \frac{\Phi}{2}. \nonumber \end{align} \tag{ 24 }$$

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This is a consequence of the level repulsion due to the off-diagonal $\pm(\Delta_L - \Delta_R)$ matrix elements coming from $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \newcommand{\hamspinsplitLR}{H_{\Delta_L,\Delta_R}} \hamspinsplitLR$. However, the degrees of degeneracy remain the same. Looking at equation (24) and at figure 3, we realise that, for positive $\Delta_L+\Delta_R$, the ground state is $\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nup_-}$, unless $|\Delta_L - \Delta_R|$ is larger than $\Delta_L + \Delta_R$, in which case the states $\newcommand{\fu}{T_{+}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fu\au}$, $\newcommand{\fd}{T_{-}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fd\au}$, $\newcommand{\fu}{T_{+}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\au\fu}$, $\newcommand{\fd}{T_{-}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\au\fd}$ (or their symmetric counterparts) span a 4-fold degenerate ground state space.

We want to briefly mention here that a generalisation of the asymmetric spin–orbit coupling which preserves the $\mathcal{T}$-symmetry, including couplings to the in-plane components of the spins, does not affect the exchange interaction. Consider

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\vec{t}}[1]{\boldsymbol{#1}} \newcommand{\su}{\uparrow} \newcommand{\ann}[1]{c_{#1}^{}} \newcommand{\cre}[1]{c_{#1}^\dagger} \displaystyle \label{eqn:generalspinsplitting} H_{\Delta_L, \Delta_R, xyz} = \sum_j \vec{t}{\Delta}_j \cdot \sum_{\tau, \sigma_1, \sigma_2} \cre{j\tau\sigma_1} (\tau_z)_{\tau\tau} (\vec{t}{\sigma})_{\sigma_1\sigma_2} \ann{j\tau\sigma_2} \nonumber \end{align} \tag{ 25 }$$

where $\newcommand{\vec{t}}[1]{\boldsymbol{#1}} \vec{t}{\Delta}_j$ is the vector of the three coupling constants $\Delta_{jx}$, $\Delta_{jy}$, $\Delta_{jz}$ for QD j, multiplied by the vector of spin Pauli matrices $\newcommand{\vec{t}}[1]{\boldsymbol{#1}} \vec{t}{\sigma}$. Equation (25) resembles the spin Zeeman term (see equation (7)), but it also preserves $\mathcal{T}$-symmetry because of the $\tau_z$ that multiplies the spin Pauli matrices. Qualitatively, a term like this should appear in the Hamiltonian of TMDC DQD systems for which it is possible to define an average local tilt and an average local curvature of the TMDC sheet for each dot [59]. However, tilt and curvature add couplings between conduction and valence band as well and one would need to check whether these couplings are small enough to be neglected. This is not the focus of this work and we only want to stress that substituting $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \newcommand{\hamspinsplitLR}{H_{\Delta_L,\Delta_R}} \hamspinsplitLR$ with $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \newcommand{\hamspinsplitgeneral}{H_{\Delta_L,\Delta_R,xyz}} \hamspinsplitgeneral$ in equation (19) does not modify the exchange interaction given in equation (23).

In this section we investigate the influence of spin and valley Zeeman terms on the symmetric spin–orbit splitting case. The total Hamiltonian for this scenario reads

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\hamvalleyzeeman}{H_V} \newcommand{\hamspinmix}{H_S} \newcommand{\hamspinsplit}{H_\Delta} \newcommand{\hamtunn}{H_t} \newcommand{\hamdetun}{H_\varepsilon} \newcommand{\hamhubb}{H_U} \newcommand{\hamtot}{H_{\rm{tot}}} \newcommand{\h}[2]{h_{#1}^{#2}} \displaystyle \label{eqn:spinmixingvalleyzeeman} \hamtot = \hamhubb + \hamdetun + \hamtunn + \hamspinsplit + \hamspinmix + \hamvalleyzeeman, \nonumber \end{align} \tag{ 26 }$$

where $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinmix}{H_S} \hamspinmix$ and $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamvalleyzeeman}{H_V} \hamvalleyzeeman$ are defined by equations (7) and (8) respectively. This is the case depicted in figure 1.

This model approximates at least two similar but distinct situations. First, when a monolayer TMDC is placed in a non-uniform and weak magnetic field, the spin of the electron in each dot couples to the local field in all three spacial directions through the spin Zeeman interaction $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinmix}{H_S} \hamspinmix$. On the other hand, here we assume that the valley DOF couples only to perpendicular magnetic field. Indeed, in [7] it was shown that such coupling exists due to orbital effects. Since in-plane magnetic field does not lead to orbital effects in the strict two-dimensional limit, we assume that it does not couple to the valley. In contrast to [7] here we only consider the lowest orbital state in each of the dots but we also take into account the Coulomb charging energy U. The other possible situation that came to our attention is that of monolayer TMDC deposited on a ferromagnetic insulator, e.g. europium oxide (EuO). Such a situation was considered in [60], where equation (7) was used to model the giant and tunable valley splitting. A Rashba term is also included, mixing conduction and valence band, that we do not consider here. Although [60] does not consider the valley Zeeman term, we note that it is allowed in the effective Hamiltonian of the TMDC because the ferromagnetic substrate breaks time-reversal symmetry. The presence of such magnetic exchange field in TMDC and ferromagnetic semiconductor heterostructures has been recently demonstrated [47, 48].

Now both $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinmix}{H_S} \hamspinmix$ and $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamvalleyzeeman}{H_V} \hamvalleyzeeman$ are non-diagonal in the basis of states presented in table 2 and they mix a large number of states. We do not list all the newly introduced matrix elements but we observe that they are confined inside the charge sectors $(0, 2)$, $(1, 1)$ and $(2, 0)$, as $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinmix}{H_S} \hamspinmix$ and $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamvalleyzeeman}{H_V} \hamvalleyzeeman$ do not contain terms of the form $\newcommand{\cre}[1]{c_{#1}^\dagger} \newcommand{\ann}[1]{c_{#1}^{}} \cre{L\tau\sigma}\ann{R\tau'\sigma'}$ (or h.c.). The only coupling between $(1, 1)$-states and $(2, 0)$ or $(0, 2)$-states remains tunneling. Nevertheless, we have to ensure that $(1, 1)$-states are lower in energy than other states. In appendix C we report the discussion of which conditions must be fulfilled in order for this to be the case. In addition, to apply the Schrieffer–Wolff transformation, equation (9) and the conditions

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle |t| \ll U \pm (\varepsilon + (h_{\ell Lz} - h_{\ell Rz})), \nonumber \end{align} \tag{ 27a }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle |t| \ll U \pm (\varepsilon - (h_{\ell Lz} - h_{\ell Rz})), \nonumber \end{align} \tag{ 27b }$$

where $\newcommand{\e}{{\rm e}} \ell=V, S$, must also be satisfied. The matrix element differences of states $\newcommand{\fd}{T_{-}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fd\au}$, $\newcommand{\fu}{T_{+}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fu\au}$, $\newcommand{\fd}{T_{-}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\au\fd}$, $\newcommand{\fu}{T_{+}} \newcommand{\au}{S} \renewcommand{\ket}[1]{|#1\rangle} \ket{\au\fu}$ with their $(2, 0)$ and $(0, 2)$ counterparts acquire a dependence on $h_{VLz} - h_{VRz}$ or $h_{SLz} - h_{SRz}$. Defining the exchange energies

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\au}{S} \newcommand{\fufd}{T_{\pm}} \newcommand{\fu}{T_{+}} \displaystyle J_{\ket{\fufd\au}} = \frac{4 |t|^2 U}{(U + \varepsilon \pm h_{VLz} \mp h_{VRz}) (U - \varepsilon \mp h_{VLz} \pm h_{VRz})}, \nonumber \end{align} \tag{ 28a }$$

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\au}{S} \newcommand{\fufd}{T_{\pm}} \newcommand{\fu}{T_{+}} \displaystyle J_{\ket{\au\fufd}} = \frac{4 |t|^2 U}{(U + \varepsilon \pm h_{SLz} \mp h_{SRz}) (U - \varepsilon \mp h_{SLz} \pm h_{SRz})}, \nonumber \end{align} \tag{ 28b }$$

the effective Hamiltonian can be written as

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\hameff}{H_{\rm{ef\,f}}} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\vec{t}}[1]{\boldsymbol{#1}} \newcommand{\nup}{n} \newcommand{\au}{S} \newcommand{\fd}{T_{-}} \newcommand{\fu}{T_{+}} \displaystyle &\hameff = -J (P_{\ket{\nup_-}} + P_{\ket{\nup_+}}) -J_{\ket{\fd\au}} P_{\ket{\fd\au}} -J_{\ket{\fu\au}} P_{\ket{\fu\au}} \nonumber \\ &\qquad -J_{\ket{\au\fd}} P_{\ket{\au\fd}} -J_{\ket{\au\fu}} P_{\ket{\au\fu}} + \Delta \Sigma \nonumber \\ &\qquad + \vec{t}{h}_{SL} \cdot \vec{t}{\sigma}_{L} + \vec{t}{h}_{SR} \cdot \vec{t}{\sigma}_{R} + h_{VLz} \tau_{Lz} + h_{VRz} \tau_{Rz}, \nonumber \end{align} \tag{ 29 }$$

where J is the standard exchange energy defined in equation (15) and $\newcommand{\vec{t}}[1]{\boldsymbol{#1}} \newcommand{\where}[2]{\left . {#1} \right |_{#2}} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinmix}{H_S} \newcommand{\e}{{\rm e}} \vec{t}{h}_{SL} \cdot \vec{t}{\sigma}_{L} + \vec{t}{h}_{SR} \cdot \vec{t}{\sigma}_{R} \equiv \where{\hamspinmix}{(1, 1)}$ while $\newcommand{\where}[2]{\left . {#1} \right |_{#2}} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamvalleyzeeman}{H_V} \newcommand{\e}{{\rm e}} h_{VLz} \tau_{Lz} + h_{VRz} \tau_{Rz} \equiv \where{\hamvalleyzeeman}{(1, 1)}$.

In the case of symmetric spin–orbit splitting ($\Delta_L = \Delta_R = \Delta$, see $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamtot}{H_{\rm{tot}}} \hamtot$ in equation (16)) there is no matrix element that allows a transition between $\mathcal{N}$ and $\mathcal{P}$. Assuming equation (9) and $2\Delta > J$ (which is usually the case for TMDCs at low detuning) we see from figure 2(c) that the subspace spanned by the states of $\mathcal{N} \times \mathcal{N}$ (also called $\mathcal{N} \times \mathcal{N}$-sector) is the LES, where,

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\sd}{\downarrow} \newcommand{\su}{\uparrow} \newcommand{\kd}{\overline{K}} \newcommand{\ku}{K} \displaystyle \mathcal{N} \times \mathcal{N} = &\left \{\ket{\ku\sd ; \ku\sd}, \quad \ket{\ku\sd ; \kd\su}, \right . \nonumber \\ & \left . \ket{\kd\su ; \ku\sd}, \quad \ket{\kd\su ; \kd\su} \right \}, \nonumber \end{align} \tag{ 30 }$$

with $\newcommand{\cre}[1]{c_{#1}^\dagger} \renewcommand{\ket}[1]{|#1\rangle} \ket{\tau_1\sigma_1 ; \tau_2\sigma_2} = \cre{L\tau_1\sigma_1} \cre{R\tau_2\sigma_2} \ket{0}$. Now we present the effective Hamiltonian when the system is restricted to this LES.

We can easily identify the negative Kramers pair $\mathcal{N}$ as a spin-1/2 DOF with

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\sd}{\downarrow} \newcommand{\su}{\uparrow} \newcommand{\kd}{\overline{K}} \newcommand{\ku}{K} \displaystyle \label{eqn:newspins} \ket{\widetilde{\su}} \equiv \ket{\kd\su}, \qquad \ket{\widetilde{\sd}} \equiv \ket{\ku\sd}, \nonumber \end{align} \tag{ 31 }$$

so that the effective basis $\mathcal{N} \times \mathcal{N}$ can be seen as the basis of all the states of two spins,

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\sd}{\downarrow} \newcommand{\su}{\uparrow} \displaystyle \mathcal{N} \times \mathcal{N} = \left \{\ket{\widetilde{\su} ; \widetilde{\su}}, \quad \ket{\widetilde{\su} ; \widetilde{\sd}}, \quad \ket{\widetilde{\sd} ; \widetilde{\su}}, \quad \ket{\widetilde{\sd} ; \widetilde{\sd}} \right \}. \nonumber \end{align} \tag{ 32 }$$

In this basis, the spin–orbit coupling always assumes the value of $-2\Delta$, which we ignore in the following since it is just an energy shift. It is well known that the effective Hamiltonian for two electrons with only the spin DOF in two QDs with only tunneling, detuning and Hubbard potential is given by [12] $\newcommand{\vec{t}}[1]{\boldsymbol{#1}} H_{\rm{ef\,f}} = J \vec{t}{S}_L \cdot \vec{t}{S}_R$, with the usual exchange energy J multiplied by the projector on the singlet state $\renewcommand{\ket}[1]{|#1\rangle} \ket{S}$, the only antisymmetric state for two spins ($\newcommand{\vec{t}}[1]{\boldsymbol{#1}} \vec{t}{S}_j$ is the vector of spin operators acting on QD j). Analogously, using the following operators,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \widetilde{\sigma}_z = \sigma_z, \qquad \widetilde{\sigma}_x = \tau_x\sigma_x, \qquad \widetilde{\sigma}_y = \tau_x\sigma_y, \nonumber \end{align} \tag{ 33 }$$

the Hamiltonian of the system restricted to basis $\mathcal{N} \times \mathcal{N}$ is

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\hameffkramers}{H_{\rm{ef\,f,}\mathcal{N} \times \mathcal{N}}} \newcommand{\hameff}{H_{\rm{ef\,f}}} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\vec{t}}[1]{\boldsymbol{#1}} \displaystyle \label{eqn:hameffhighsymmsplit} \hameffkramers = J \widetilde{\vec{t}{S}}_L \cdot \widetilde{\vec{t}{S}}_R, \nonumber \end{align} \tag{ 34 }$$

where $\widetilde{S}_{ji} = \frac{1}{2}\widetilde{\sigma}_{ji}$ is the spin operator proportional to the new Pauli operator $\widetilde{\sigma}_i$, $i = x, y, z$, acting on QD $j = L, R$. The ground state space is one-dimensional as for the full $(1, 1)$-subspace. Indeed, the ground state is the same,

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\nup}{n} \newcommand{\sd}{\downarrow} \newcommand{\su}{\uparrow} \newcommand{\kd}{\overline{K}} \newcommand{\ku}{K} \displaystyle \nonumber \ket{\widetilde{S}} =& \left (\ket{\widetilde{\su} ; \widetilde{\sd}} - \ket{\widetilde{\sd} ; \widetilde{\su}} \right) / \sqrt{2} \nonumber \\ =& \left (\ket{\ku \sd ; \kd \su} - \ket{\kd \su ; \ku \sd} \right) / \sqrt{2} = \ket{\nup_-}. \nonumber \end{align} \tag{ 35 }$$

Very similar considerations are valid when we relax the constraint of equal spin–orbit splittings in the dots ($\Delta_L \neq \Delta_R$, see $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamtot}{H_{\rm{tot}}} \hamtot$ in (19)). To reduce the LES to the subspace spanned by $\mathcal{N} \times \mathcal{N}$ we need

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta_L + \Delta_R > (J+\sqrt{J^2 + 4(\Delta_L - \Delta_R)^2})/2, \nonumber \end{align} \tag{ 36 }$$

see figure 3 and equation (24). Of course, to apply perturbation theory, the conditions $|t| \ll | U \pm (\varepsilon + \Delta_L - \Delta_R) |$ and $|t| \ll | U \pm (\varepsilon -$$\Delta_L + \Delta_R) |$ must be valid here too.

Following the same steps that led to equation (34), we arrive to the effective Hamiltonian,

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\hameffkramers}{H_{\rm{ef\,f,}\mathcal{N} \times \mathcal{N}}} \newcommand{\hameff}{H_{\rm{ef\!f}}} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\vec{t}}[1]{\boldsymbol{#1}} \newcommand{\nup}{n} \displaystyle \label{eqn:heffkramersasymm} \hameffkramers = J_{\ket{\nup_-}} \widetilde{\vec{t}{S}}_L \cdot \widetilde{\vec{t}{S}}_R \nonumber \end{align} \tag{ 37 }$$

where $\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} J_{\ket{\nup_-}}$ is given in equation (22). Indeed, from figure 3 we see that the energy levels for the $\mathcal{N} \times \mathcal{N}$-sector are similar between symmetric and asymmetric spin–orbit splitting, only the exchange energy J is altered.

To better understand what is the effect of $\Delta_L - \Delta_R$ in $\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} J_{\ket{\nup_-}}$ with respect to J, we may write

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\nup}{n} \displaystyle J_{\ket{\nup_-}} = \frac{J}{\left (1 + \frac{\Delta_R - \Delta_L}{U + \varepsilon} \right) \left (1 + \frac{\Delta_L - \Delta_R}{U - \varepsilon} \right)}. \nonumber \end{align} \tag{ 38 }$$

Since $(\Delta_L - \Delta_R)/(U \pm \varepsilon) \ll 1$, we can write

$$\begin{align} \renewcommand{\ket}[1]{|#1\rangle} \newcommand{\e}{{\rm e}} \newcommand{\nup}{n} \displaystyle J_{\ket{\nup_-}} \simeq J \left (1 - 2 \varepsilon \frac{\Delta_L - \Delta_R}{U^2 - \varepsilon^2} \right). \nonumber \end{align} \tag{ 39 }$$

The effect of an almost symmetric spin–orbit coupling on the exchange energy is that of fine tuning around the value of J by a small positive or negative quantity, depending on the sign of ε and $\Delta_L - \Delta_R$. Moreover, as $\varepsilon \to 0$, $\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} J_{\ket{\nup_-}} \to J$. The above results remain valid for the generalisation to $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \newcommand{\hamspinsplitgeneral}{H_{\Delta_L,\Delta_R,xyz}} \hamspinsplitgeneral$ (equation (25)).

We now consider the total Hamiltonian of equation (26) restricted to the $\mathcal{N} \times \mathcal{N}$-sector. The results in this section are valid when the spin and valley Zeeman coupling constants are weak compared to the spin–orbit strength: $|h_{Sji}|, |h_{Vjz}| \ll 2\Delta$, $j = L, R$, $i = x, y, z$. We show directly and discuss the effective Hamiltonian for the subspace spanned by $\mathcal{N} \times \mathcal{N}$ as a $4 \times 4$ matrix. The columns are associated, from left to right, to the rotated basis states $\newcommand{\nup}{n} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nup_-}$, $\newcommand{\nd}{\overline{n}} \renewcommand{\ket}[1]{|#1\rangle} \ket{\nd_-}$, $\newcommand{\fu}{T_{+}} \newcommand{\fd}{T_{-}} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fu\fd}$ and $\newcommand{\fu}{T_{+}} \newcommand{\fd}{T_{-}} \renewcommand{\ket}[1]{|#1\rangle} \ket{\fd\fu}$,

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\hameffkramers}{H_{\rm{ef\,f,}\mathcal{N} \times \mathcal{N}}} \newcommand{\hameff}{H_{\rm{ef\,f}}} \newcommand{\HV}[1]{H_V^{#1}} \newcommand{\HS}[1]{H_S^{#1}} \newcommand{\dhV}[1]{\delta h_V^{#1}} \newcommand{\dhS}[1]{\delta h_S^{#1}} \newcommand{\h}[2]{h_{#1}^{#2}} \displaystyle \label{eqn:heffkramersspinmixingvalleyzeeman} \hameffkramers = \left(\begin{array}{@{}ccccc@{}} -J - A & \dhS{z} - \dhV{z} & 0 & 0 \nonumber \\ \dhS{z} - \dhV{z} & -A & 0 & 0 \nonumber \\ 0 & 0 & - \HS{z} + \HV{z} - A_{+} & 0 \nonumber \\ 0 & 0 & 0 & \HS{z} - \HV{z} - A_{-} \end{array}\right), \nonumber \end{align} \tag{ 40 }$$

where J is the standard exchange energy as in equation (15). We use the notation

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\Hl}[1]{H_\ell^{#1}} \newcommand{\h}[2]{h_{#1}^{#2}} \displaystyle \Hl{z} = \h{\ell Lz}{} + \h{\ell Rz}{}, \nonumber \end{align} \tag{ 41a }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\dhl}[1]{\delta h_\ell^{#1}} \newcommand{\h}[2]{h_{#1}^{#2}} \displaystyle \dhl{z} = \h{\ell Lz}{} - \h{\ell Rz}{}, \nonumber \end{align} \tag{ 41b }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\h}[2]{h_{#1}^{#2}} \displaystyle \h{Sj}{\pm} = \h{Sjx}{} \pm i \h{Sjy}{}, \nonumber \end{align} \tag{ 41c }$$

with $\newcommand{\e}{{\rm e}} \ell = S, V$, $j=L, R$ and $A_{\pm}$ and A are given by

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\HS}[1]{H_S^{#1}} \newcommand{\h}[2]{h_{#1}^{#2}} \displaystyle A_{\pm} = \frac{\h{SL}{+}\h{SL}{-} + \h{SR}{+}\h{SR}{-}}{2\Delta \pm \HS{z}}, \nonumber \end{align} \tag{ 42a }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\HS}[1]{H_S^{#1}} \newcommand{\h}[2]{h_{#1}^{#2}} \displaystyle A = \frac{A_{-} + A_{+}}{2} = 2\Delta \frac{\h{SL}{+}\h{SL}{-} + \h{SR}{+}\h{SR}{-}}{4\Delta^2 - {(\HS{z})}^2}. \nonumber \end{align} \tag{ 42b }$$

Since $|h_{Sjz}| \ll 2\Delta$, we can use the approximation $A_{+} \approx A_{-} \approx A$, thus the contribution of A to equation (40) is an energy shift that can be ignored. Therefore, in this case we can also write

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\hamvalleyzeeman}{H_V} \newcommand{\hamspinmix}{H_S} \newcommand{\hameffkramers}{H_{\rm{ef\,f,}\mathcal{N} \times \mathcal{N}}} \newcommand{\hameff}{H_{\rm{ef\,\!f}}} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\where}[2]{\left . {#1} \right |_{#2}} \newcommand{\vec{t}}[1]{\boldsymbol{#1}} \displaystyle \hameffkramers = J \widetilde{\vec{t}{S}}_L \cdot \widetilde{\vec{t}{S}}_R + \where{\hamspinmix}{\mathcal{N} \times \mathcal{N}} + \where{\hamvalleyzeeman}{\mathcal{N} \times \mathcal{N}}, \nonumber \end{align} \tag{ 43 }$$

where $\newcommand{\where}[2]{\left . {#1} \right |_{#2}} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinmix}{H_S} \where{\hamspinmix}{\mathcal{N} \times \mathcal{N}}$ and $\newcommand{\where}[2]{\left . {#1} \right |_{#2}} \newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamvalleyzeeman}{H_V} \where{\hamvalleyzeeman}{\mathcal{N} \times \mathcal{N}}$ are the restrictions of $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinmix}{H_S} \hamspinmix$ and $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamvalleyzeeman}{H_V} \hamvalleyzeeman$ to the $\mathcal{N} \times \mathcal{N}$-sector.

Given the generality of $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \newcommand{\hamspinsplitgeneral}{H_{\Delta_L,\Delta_R,xyz}} \hamspinsplitgeneral$ (equation (25)) as a $\mathcal{T}$-symmetric term and the fact that $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinsplit}{H_\Delta} \newcommand{\hamspinsplitgeneral}{H_{\Delta_L,\Delta_R,xyz}} \hamspinsplitgeneral$ and $\newcommand{\h}[2]{h_{#1}^{#2}} \newcommand{\hamspinmix}{H_S} \hamspinmix$ only differ in the presence of $\tau_z$ that multiplies the spin Pauli matrices, we can assert that the new matrix elements ($\newcommand{\HS}[1]{H_S^{#1}} \HS{z}$, $\newcommand{\HV}[1]{H_V^{#1}} \HV{z}$, $\newcommand{\dhS}[1]{\delta h_S^{#1}} \dhS{z}$, $\newcommand{\dhV}[1]{\delta h_V^{#1}} \dhV{z}$, $A_{\pm}$ and A) that appear in equation (40) as compared to equation (34), are allowed only by breaking the time-reversal symmetry.