Probing ground-state degeneracies of a strongly interacting Fermi-Hubbard model with superconducting correlations

Abstract

The Fermi-Hubbard model and its rich phase diagram naturally emerges as a description for a wide range of electronic systems. Recent advances in semiconductor-superconductor hybrid quantum dot arrays have allowed to realize degenerate quantum systems in a controllable way, e.g., allowing to observe robust zero-bias peaks in Kitaev chains, indicative for Majorana bound states. In this work, we connect these two domains. Noting the strong on-site Coulomb repulsion within quantum dots, we study small arrays of spinful hybrid quantum dots implemented in a two-dimensional electron gas. This system constitutes a Fermi-Hubbard model with inter-site superconducting correlations. For two electronic sites, we find robust zero-bias peaks indicative of a strongly degenerate spectrum hosting emergent Majorana Kramers pairs or $\mathbb{Z}_3$-parafermions. Extending to three sites, we find that these spinful systems scale very differently compared to spinless Kitaev chains. When the sweet-spot conditions are satisfied pairwise, we find that the ground state degeneracy of the full three-site system is lifted. This degeneracy can be restored by tuning the superconducting phase difference between the hybrid segments. However, these states are not robust to quantum dot detuning. Our observations are a first step towards studying degeneracies in strongly interacting Fermi-Hubbard systems with superconducting correlations.

Figures (14)

• Figure 1: Model and device schematics(a) Schematic of the nearest-neighbour interactions for two spinful fermionic sites coupled via a superconductor. We assume equal tunneling rates for spin-up and spin-down such that $t^{(i,j)}=t^{(i,j)}_{\downarrow\downarrow}=t^{(i,j)}_{\uparrow\uparrow}$ and $\Delta^{(i,j)}=\Delta^{(i,j)}_{\uparrow\downarrow}=-\Delta^{(i,j)}_{\downarrow\uparrow}$, where the sign is due to time reversal symmetry. To implement the $U=\infty$ limit, the double occupied state is disallowed. (b) False colour scanning electron micrograph of a copy of device A. Two aluminium strips are connected via a continuous loop with a radius of 10µm (not drawn to scale). Scale bar is 500nm. (c) Schematic representation of the three site device, highlighting the parameters relevant for the theoretical simulations.
• Figure 2: The two-site sweet spot. Tuning the coupling between QD1 and QD2, with QD3 tuned to a Coulomb blockaded regime. (a) Exemplar CSD in an ECT dominated regime ($V_{\mathrm{ABS}}^{\mathrm{(1)}}$ = -190mV), showing four anti-diagonal avoided crossings. (b) Exemplar CSD in a CAR dominated regime ($V_{\mathrm{ABS}}^{\mathrm{(1)}}$ = -137.5mV), showing four diagonal avoided crossings. Examples were chosen to showcase the change in connectivity can arise in all 4 quadrants of the CSDs. More general CSDs are shown in \ref{['fig:S3_extended_CSDs_devA']}. (c) Close up CSD of the bottom left quadrant, at an intermediate value of $V_{\mathrm{ABS}}^{\mathrm{(1)}}$ = -180.3mV. (d) Energy level diagram for the two parity sectors at the sweet spot (detailed in the main text). Coherence factors have been omitted for brevity. (e) Tunnelling spectroscopy measured when detuning $V_{\mathrm{QD1}}$, with $V_{\mathrm{QD2}}$ set on resonance. Coloured arrows correspond to transitions indicated in (d). Reproduction of results in a separate device is shown in \ref{['fig:S5_devBcharacterisation']}-\ref{['fig:S7_reproduce_spectra_devB']}.
• Figure 3: Scaling from two to three sites. Measurements obtained with both pairs of QDs individually tuned to a sweet spot configuration. (a) Spectroscopy of the left ($S_{\mathrm{1}}$) and right ($S_{\mathrm{3}}$) while sweeping $V_{\mathrm{QD3}}$. Once QD3 is on resonance, the ZBP on both sides is observed to split. (b) Numerically calculated conductance with (settings), showing similar behaviour. To understand this, we consider schematically the situation when QD3 is (c) off and (d) on resonance. When QD3 is off resonance, the triplet subspace is uncoupled from the rest of the system and a pair of spatially overlapping Majorana Kramers pairs is present on both QDs. Adding the third QD couples to both zero energy modes and allows the triplet states to participate, favouring the even ground-state in energy (see Ref. bozkurt2025 for further details).
• Figure 4: Superconducting phase as tuning knob. To understand where zero-energy modes arise in the full three-site system, a larger parameter space is explored (procedure detailed in \ref{['fig:S8_full_data_figure_4']}). For each QD pair, sweet spot values are obtained, denoted as $\delta$$V_{\mathrm{ABS}}^{\mathrm{(i)}}$ = 0. Zero-bias conductance measurements are performed upon simultaneously varying $\delta V_{\mathrm{QDi}}$ versus sweeping $V_{\mathrm{ABS}}^{\mathrm{(i)}}$ around their sweet-spots. (a) Measurement obtained for $B_z$ = 4µT, corresponding roughly to $\phi$ = 0. (b) We compare this to a numerical simulation, where all $\mu$ and all $t$ parameters are swept together, for fixed $\Delta$. (c-f) We repeat the measurements and simulations for $B_z$ = 15µT (c), corresponding rouhgly to $\phi$ = $\pi$ (d) and (e) $B_z$ = 11µT corresponding to (f) $\phi$$\approx$ 1.82.
• Figure 5: Strongly degenerate ground states in the three-site chain.(a) Analytical solution yielding $E_{\mathrm{gs}}^{\mathrm{odd}}=E_{\mathrm{gs}}^{\mathrm{even}}$ for the three-site chain when $\mu_i=0$ (given by Eq. \ref{['eq:magic_line']}), resulting in a strongly degenerate spectrum. (b) Derivative of the energy difference between the odd and even ground states with respect to detuning $\mu$. The derivatives vanish at two specific points. The significance can be seen when comparing the conductance spectrum of detuning $\mu_{\mathrm{3}}$ at (c)$\phi$ = $\pi$ and (d)$\phi$ = 1.861. The numerical energy spectrum is overlaid in (d) to highlight that a small energy splitting still arises at larger detuning of $\mu_3$. (e) Perturbative expansion of first order and second order derivatives, highlighting that the higher order derivative does not vanish at the same $\phi$ value as the first order. (f) Measurement of $\tilde{S}_{\mathrm{1}}$ when detuning QDR, with the settings in \ref{['fig:Fig4']}c, in order to compare with the predicted spectrum (d). Repeated measurements for a range of $B_{\mathrm{x}}$ values is shown in \ref{['fig:S9_detuning_vs_phase']}.