Negative exchange interaction in Si quantum dot arrays via valley-phase induced $\mathbb{Z}_2$ gauge field

TL;DR

The paper shows that valley-phase differences in Si quantum dot arrays generate a $\mathbb{Z}_2$ gauge field on interdot links, enabling a negative exchange $J<0$ in the two-electron sector without an external magnetic field. By deriving a low-energy ground-valley Hamiltonian with $t'_{i,j} = t_{i,j} cos(phi_{i,j}/2)$ and introducing a gauge field on links, the authors connect valley physics to $\pi$-fluxes threading plaquettes, which modify exchange and can even break Nagaoka ferromagnetism. They explore triangular and square plaquettes to demonstrate negative exchange and flux-driven ferromagnetism breaking, and show that valley-phase configurations can be engineered on demand by selectively occupying excited valleys or using ancillary dots, enabling programmable exchange across 2D dot arrays and 1D chains. The work suggests a path to dynamically corrected exchange-based quantum gates and provides a framework for studying $\mathbb{Z}_2$ gauge-field effects and valley-driven quantum magnetism in semiconductor quantum-dot systems.

Abstract

The exchange interaction $J$ offers a powerful tool for quantum computation based on semiconductor spin qubits. However, the exchange interaction in two-electron systems in the absence of a magnetic field is usually constrained to be non-negative $J \geq 0$, which inhibits the construction of various dynamically corrected exchange-based gates. In this work, we show that negative exchange $J < 0$ can be realized in two-electron Si quantum dot arrays in the absence of a magnetic field due to the presence of the valley degree of freedom. Here, valley phase differences between dots produce a non-trivial $\mathbb{Z}_2$ gauge field in the low-energy effective theory, which in turn can lead to a negative exchange interaction. In addition, we show that this $\mathbb{Z}_2$ gauge field can break Nagaoka ferromagnetism and be engineered by altering the occupancy of the dot array. Therefore, our work uncovers new tools for exchange-based quantum computing and a novel setting for studying quantum magnetism.

Figures (7)

• Figure 1: A quantum dot array, where each dot has both spin $\sigma \!\in\! \{\uparrow,\downarrow\}$ and valley $\tau \in \{+,-\}$ degrees of freedom, as indicated by the four colored circles in the blown up dot at the right. A black line connecting dots $i$ and $j$ denotes tunnel coupling $t_{i,j}$ that preserves spin and valley. Here, $t_{i,j} < 0$ due to the s-wave symmetry of each dot's lowest-energy orbital. The valleys in dot $i$ are coupled by $\Delta_i = |\Delta_i|e^{i\phi_i}$, where $E_{v,i} = 2|\Delta_i|$ is the valley splitting and $\phi_i$ is the valley phase. The relative valley phases between dots plays a key role in the low-energy physics.
• Figure 2: Energy level diagram of two coupled dots after transforming into the ground and excited valley basis. The ground and excited valleys of each dot are separated in energy by the valley splitting $E_{v,i} = 2|\Delta_i|$. There exists both intra-valley and inter-valley tunnel couplings with both the magnitudes and signs being determined by the valley phase difference $\phi_{2,1} = \phi_2 - \phi_1$ between the dots.
• Figure 3: (a) Effective quantum dot array after projecting onto the ground valley of each dot. Each dot only a spin degree of freedom, as indicated by the two circles in the blown up dot. Here, the two colors of each circle indicates that the ground valley is an equal superpositions of the + and - valleys shown in Fig. \ref{['FIGqdArray']}. The sign of the effective tunnel coupling $t^\prime_{i,j}$ is determined by the valley phase difference $\phi_{i,j} = \phi_{i} - \phi_{j}$, where $t_{i,j}^\prime > 0$ ($t_{i,j}^\prime< 0$) are indicated by red (black) lines. The sign of the effective tunnel coupling defines a $\mathbb{Z}_2$ gauge field on each link between dots, $\chi_{i,j} = \text{sgn}(t_{i,j}^\prime) = \pm 1$. Plaquettes with an odd number of $t_{i,j}^\prime > 0$ tunnel couplings have a gauge-invariant $\pi$-flux, which is equivalent to a (superconducting) magnetic flux quantum $\Phi_0$ threading through the plaquette. Such $\pi$-fluxes can lead to interesting phenomena, such as negative exchange interactions and broken Nagaoka ferromagnetism, as demonstrated below. (b) System in (a) after performing the gauge transformation indicated by the $\pm$ factors near each dot. Notice that the $\mathbb{Z}_2$ gauge field configuration changes, but the $\mathbb{Z}_2$ flux configuration is invariant under a gauge transformation.
• Figure 4: (a) Triangular quantum dot plaquette. Each dot has an inter-valley coupling $\Delta_i = |\Delta_{i}|e^{i\phi_i}$, where $\phi_{i}$ is the valley phase. Solid black lines indicate inter-dot tunnel couplings $t_{i,j} \leq 0$. (b) Singlet-triplet splitting $E_{\text{ST}}$ for $M = 2$ electrons in a triangular plaquette as a function of $\phi_2$ and $\phi_3$. Without loss of generality, we set $\phi_1 = 0$. Other parameters are $t_{i,j} = t < 0$ for all $i,j$, $|\Delta_{i}| = 50|t|$ and $\varepsilon_{i}^\prime = \varepsilon_i - |\Delta_i| = 0$ for all $i$, and $U = 1000 |t|$. $E_{\text{ST}} < 0$ (i.e. a negative exchange interaction $J < 0$) is realized in the blue regions, covering $1/4$ of the valley phase parameter space. For these regions, $t_{3,2}^\prime\! > \! 0$ ($\chi_{3,2} = 1$), yielding a $\pi$-flux threading the plaquette in the low-energy theory given in Eq. (\ref{['Hgv']}). (c) Square plaquette. (d) Energy splitting between the lowest-energy $S = 1/2$ and $S = 3/2$ states for $M = 3$ electrons in a square plaquette as a function of $\phi_2$ and $\phi_3$. $\phi_1 = 0$ without loss of generality, and we set $\phi_4 = \pi/2$. Other parameters are the same as (b). In the absence of valley phase difference, the square plaquette exhibits Nagaoka ferromagnetism ($S = 3/2$ ground state). A $\pi$-flux breaks the Nagaoka ferromagnetism, leading to a $S = 1/2$ ground state, as demonstrated by the blue regions, where $E_{1/2} - E_{3/2} < 0$. Note that for $M = 2$ electrons in a square plaquette, $E_{ST} < 0$ will be realized in the same regions of valley phase parameter space that have $E_{1/2} - E_{3/2} < 0$ in (d).
• Figure 5: (a) Third-order processes leading to the exchange of two electrons occupying the ground valleys of the lowers dots. The upper and lower branches represent example processes involving the ground and excited valleys, respectively, of dot 3. The relative contributions of the two processes depends on the ratio of the third dot's valley splitting $2 |\Delta_3|$ and the inter-dot detuning $\varepsilon_3^\prime - \varepsilon_{1}^\prime = \varepsilon_3^\prime - \varepsilon_{2}^\prime$. (b) Singlet-triplet splitting $E_{ST}$ as a function of $\varepsilon_3^\prime$ and $|\Delta_3|$. Other parameters are $\phi_1 = 0$, $\phi_{2} = 2\pi/3$, $\phi_{3} = -2\pi/3$, $t_{i,j} = -|t| < 0$ for all $i,j$, $|\Delta_1| = |\Delta_2| = 50|t|$, and $U = 1000 |t|$. While $E_{ST} < 0$ in the limit of large $|\Delta_3|$, as consistent with Fig. \ref{['FIGresults1']}(b), sufficiently small $|\Delta_{3}|$ leads to $E_{ST} > 0$. In the $E_{ST} > 0$ region, the contribution of the third-order processes involving the excited valley of dot $3$ are counteracting and larger than the negative-exchange third-order processes involving the ground valley of dot $3$.