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# Quantum simulation of an exotic quantum critical point in a two-site charge Kondo circuit

## Abstract

Tuning a material to the cusp between two distinct ground states can produce physical properties that are unlike those in either of the neighbouring phases. Advances in the fabrication and control of quantum systems have raised the tantalizing prospect of artificial quantum simulators that can capture such behaviour. A tunable array of coupled qubits should have an appropriately rich phase diagram, but realizing such a system with either tunnel-coupled semiconductor quantum dots or metal nanostructures has proven difficult. The challenge for scaling up to clusters or lattices is to ensure that each element behaves essentially identically and that the coupling between elements is uniform, while also maintaining tunability of the interactions. Here we study a nanoelectronic circuit comprising two coupled hybrid metal–semiconductor islands, combining the strengths of both materials to form a potentially scalable platform. The semiconductor component allows for controllable inter-site couplings at quantum point contacts, while the metal component’s effective continuum of states means that different sites can be made equivalent by tuning local potentials. The couplings afforded by this architecture can realize an unusual quantum critical point resulting from frustrated Kondo interactions. The observed critical behaviour matches theoretical predictions, verifying the success of our experimental quantum simulation.

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## Data availability

All data used in this work are available in the Stanford Digital Repository at https://doi.org/10.25740/mx151nn9365. Source data are provided with this Paper.

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## Acknowledgements

We thank F. Pierre, I. Safi, G. Zarand, C. P. Moca, I. Weymann, P. Sriram, E. Sela, Y. Oreg, Q. Si, C. Varma and C. Mora for their scientific insights and suggestions. To make this project work, before coupling two islands we had to start by reproducing F. Pierre’s tour de force experiments on single islands of the same type. F. Pierre helped with comments on our fabrication process, measurement procedure and analysis. Discussions with C. Mora led to improvements in our analysis of the periodic Fermi-liquid scale. We acknowledge G. Zarand, C. P. Moca and I. Weymann for early discussions of the Hamiltonian and its implications. Measurement and analysis were supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences (BES), under contract no. DE-AC02-76SF00515. Growth and characterization of heterostructures was supported by the French Renatech network. Theory and computation (A.K.M.) were supported by the Irish Research Council Laureate Awards 2017/2018 through grant no. IRCLA/2017/169. Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under award ECCS-2026822. Early research that established how to meet the demanding technical conditions for sample fabrication and basic measurements was supported by the National Science Foundation (NSF) under award no. 1608962. W.P. acknowledges support from the Fletcher Jones Fellowship. C.L.H. acknowledges support from the Gabilan Fellowship. L.P. acknowledges support of the Albion Walter Hewlett Fellowship.

## Author information

Authors

### Contributions

W.P. and L.P. performed the measurements. L.P. fabricated the device. A.K.M. developed the theory and carried out NRG calculations. W.P., L.P., C.L.H., M.A.K., A.K.M. and D.G.-G. analysed the data. A.C. and U.G. grew the heterostructure that hosts the 2DEG on which these samples are built. D.G.-G. supervised the project.

### Corresponding authors

Correspondence to Andrew K. Mitchell or David Goldhaber-Gordon.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

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Nature Physics thanks Takis Kontos and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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## Extended data

### Extended Data Fig. 1 Schematic illustrations of the models discussed.

a, The DCK model consists of six effectively independent spinless conduction electron reservoirs (blue for island, red for lead), described by fermionic operators fαγ and cα for α = L, R and γ = l, i. Tunneling occurs at each of the three QPCs controlled by JL,C,R. The island charging energy $${E}_{C}^{\alpha }$$ correlates electrons fαi and cα on the same island either side of the metal component (black bar). b, At a TP, Eq. (9) describes the system at low temperatures. The three retained charge states of the two-island structure (denoted $$\left\vert A\right\rangle ,\left\vert B\right\rangle ,\left\vert C\right\rangle$$) are interconverted by QPC tunneling. The frustrated QCP arises when JL = JR = JC ≡ J. The conductive pathway $$\left\vert A\right\rangle \to \left\vert B\right\rangle \to \left\vert C\right\rangle$$ corresponds to transport from left lead to right lead, and is illustrated with the blue arrows (the flow is reversed $$\left\vert C\right\rangle \to \left\vert B\right\rangle \to \left\vert A\right\rangle$$ by changing the sign of the applied bias voltage).

### Extended Data Fig. 2

SEM micrograph of nominally identical device. The acceleration voltage in the SEM was 5 kV.

### Extended Data Fig. 3 Dynamical Coulomb blockade of QPC transmissions.

a. Measured QPC transmissions τR, τL as a function of a source-drain bias VSD for different QPC gate voltages. The measured transmission is extracted by measuring the series conductance when in series with the inter-island QPC and opposite island-lead QPC set to fully transmit a single channel (τC,L/R = 1). The measured transmissions clearly dip at zero bias, consistent with dynamical Coulomb blockade (DCB) behavior. The high bias behavior (VSD ≈ 50μV) recovers the ‘intrinsic’ transmission of the QPC, unrenormalized by DCB. b. DCB measurements comparing the right island-lead QPC to the inter-island QPC. It is clear there is a substantial difference in the DCB-renormalization at zero bias between the two, likely due to the device geometry. c. Comparison of measuring τR through both islands (blue lines, as in a, b) and through the adjacent plunger gate PR (red lines). While typically we would expect no significant bias dependence when measuring through PR, we in fact see DCB-like behavior. d. Comparing the two measurement pathways of c at fixed source-drain bias as a function of the QPC gate voltage. The ‘through the island’ (blue) measurements have been shifted by 9mV to account for the large capacitive cross-talk effect when switching between the two different measurement pathways. That the high bias traces match well is indicative that there is indeed DCB-renormalization of the transmission when measuring through PR. Empirically, using the zero bias, ‘through the plunger gate’ measurement of the transmission (solid red line), best captures the relevant transmissions in the Kondo interactions of our system.

### Extended Data Fig. 4 Semi-universal τC values.

a, Measured inter-island transmission as a function of an applied gate voltage. The markers correspond to the inter-island transmissions used in Fig. 3a of the main text. b, Original line cuts in which the truncated data used in Fig. 3a are extracted from.

## Supplementary information

### Supplementary Information

Supplementary discussion and Figs. 1–7

### Supplementary Data 1

Statistical source data.

### Supplementary Data 2

Statistical source data.

### Supplementary Data 3

Statistical source data.

### Supplementary Data 4

Statistical source data.

### Supplementary Data 5

Statistical source data.

### Supplementary Data 6

Statistical source data.

### Supplementary Data 7

Statistical source data.

## Source data

### Source Data Fig. 1

Statistical source data.

### Source Data Fig. 2

Statistical source data.

### Source Data Fig. 3

Statistical source data.

### Source Data Extended Data Fig./Table 3

Statistical source data.

### Source Data Extended Data Fig./Table 4

Statistical source data.

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Pouse, W., Peeters, L., Hsueh, C.L. et al. Quantum simulation of an exotic quantum critical point in a two-site charge Kondo circuit. Nat. Phys. (2023). https://doi.org/10.1038/s41567-022-01905-4

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• DOI: https://doi.org/10.1038/s41567-022-01905-4