Tunable quantum criticalities in an isospin extended Hubbard model simulator

Abstract

Studying strong electron correlations has been an essential driving force for pushing the frontiers of condensed matter physics. In particular, in the vicinity of correlation-driven quantum phase transitions (QPTs), quantum critical fluctuations of multiple degrees of freedom facilitate exotic many-body states and quantum critical behaviours beyond Landau's framework¹. Recently, moiré heterostructures of van der Waals materials have been demonstrated as highly tunable quantum platforms for exploring fascinating, strongly correlated quantum physics^2-22. Here we report the observation of tunable quantum criticalities in an experimental simulator of the extended Hubbard model with spin-valley isospins arising in chiral-stacked twisted double bilayer graphene (cTDBG). Scaling analysis shows a quantum two-stage criticality manifesting two distinct quantum critical points as the generalized Wigner crystal transits to a Fermi liquid by varying the displacement field, suggesting the emergence of a critical intermediate phase. The quantum two-stage criticality evolves into a quantum pseudo criticality as a high parallel magnetic field is applied. In such a pseudo criticality, we find that the quantum critical scaling is only valid above a critical temperature, indicating a weak first-order QPT therein. Our results demonstrate a highly tunable solid-state simulator with intricate interplay of multiple degrees of freedom for exploring exotic quantum critical states and behaviours.

Fig. 1. cTDBG with θ = 0.75°.

a, Schematic of our cTDBG device. Bottom and top gate V_bg and V_tg are adopted to independently control carrier density n and displacement field D. b, Moiré patterns of small-twist-angle cTDBG. Three regions of different stacking orders are indicated by circles of different colours. c, Top and side views of the different stacked regions marked in b. d, Longitudinal resistance (R) and Hall resistance (R_H) as functions of n measured at D = 0 V nm⁻¹, $B_{\perp }=200\mathrm{mT}$ and T = 1.5 K. e, 2D map of R as a function of n and D. f, Calculated band structures for different displacement fields D. The Chern numbers are labelled for the first and second conduction bands.

Fig. 2. Evidences for Wigner crystal state.

a, High-resolution 2D plot of R. Correlated insulating features emerge in the second moiré band. b, Line plots extracted from the dashed lines in a for selected displacement fields. The dashed arrows indicate peaks at n = 7n₀ and $7\frac{2}{3}{n}_0$. c,d, Differential resistance versus d.c. bias current at various displacement fields and temperatures, respectively. e, Resistance in log scale versus $T^{-\frac{1}{2}}$ at selected $B_{\parallel }$. Inset, resistance in log scale as a function of $T^{-0.7}$ at large $B_{\parallel }$. f,g, 2D map of R as a function of the filling factor and parallel (perpendicular) magnetic field measured at D = −0.46 V nm⁻¹.

Fig. 3. Quantum two-stage criticality.

a, Line plots of the resistance for a series of D (R–T traces are offset for better clarification). b,c, 2D maps of R and dR/dT as functions of D and T. Orange and red markers represent the phase boundaries, with error bars defined in Methods. D_c and D_n are the two quantum critical points. d,e, Scaling analysis for the quantum two-stage criticality in the Wigner crystal regime and the normal metal regime, respectively, with temperature scaling parameter T₀ chosen to yield collapse of the R–T curves. Insets, T₀ versus |D − D_c| and |D − D_n|, which follow power-law behaviours.

Fig. 4. Quantum pseudo criticality in a 12-T parallel magnetic field.

a, Line plots of R–T curves at different D (the curves are offset for better clarification). The dashed box indicates the almost-flat R–T curves. b, 2D map of dR/dT as a function of D and T for $B_{\parallel }=12\mathrm{T}$. c, Successful (lower inset) and failed (upper inset) collapse of R–T curves by performing the scaling analysis in the Wigner crystal regime for temperatures above T* only and for the full temperature range, indicating the emergence of a quantum pseudo criticality. d, Schematic of phase diagram and quantum phase transitions in the plane of $D-B_{\parallel }$.

Extended Data Fig. 1. Parallel magnetic field dependence of resistance at different D.

Resistance as a function of parallel magnetic field at selected displacement fields. The saturation of resistance at high parallel magnetic field suggests that the spins of the localized electrons are totally polarized.

Extended Data Fig. 2. Line traces extracted from Fig. 2f,g at several magnetic fields.

a, As parallel magnetic field increases, resistance at fractional filling $n=7\frac{2}{3}{n}_0$ first increases monotonically and gradually saturates. b, For the case of vertical magnetic field, resistance at fractional filling $n=7\frac{2}{3}{n}_0$ increases first and is finally overwhelmed by quantum Hall effect at high $B_{\perp }$.

Extended Data Fig. 3. Failure of scaling analysis for Wigner crystal and normal metal by choosing D_c and D_n within the intermediate regime.

Here we show two typical failures of scaling. a, Scaling analysis for Wigner crystal by choosing D_c = −0.316 V nm⁻¹. The curves are unable to collapse onto a single curve compared with the scaling analysis in the main text with D_c = −0.325 V nm⁻¹. b, Scaling analysis is unable to collapse the R–T curves if we choose D_n = −0.305 V nm⁻¹ as the critical displacement field for the normal metal regime.

Extended Data Fig. 4. Evolution of the quantum critical behaviour with parallel magnetic field.

2D map of dR/dT as a function of D and T at $B_{\parallel }=1.5\mathrm{T}$ (a) and 3 T (b).

Extended Data Fig. 5. Quantum pseudo criticality in the normal metal regime.

Successful (a) and failed (b) collapse of scaled R–T curves in the normal metal regime for temperatures above T* only and for the full temperature range.

Extended Data Fig. 6. r_s evolution with varying filling factor and Fermi surfaces at n=723n0 at different D.

r_s of AB-BA-stacked graphene under D = −0.4 V nm⁻¹ from hole pockets near Γ_s points (a) and D = 0 V nm⁻¹ from hole pockets near M_s points (red), hole pockets near K_s points (orange) and electron pockets near Γ_s points (green) (c). Fermi surfaces at the filling factor $n=7\frac{2}{3}{n}_0$ at D = −0.4 V nm⁻¹ (b) and D = 0 V nm⁻¹ (d). The blue (red) line represents the pockets from the K (K′) valley. The solid black line represents the moiré Brillouin zone.

Extended Data Fig. 7. Determination of T_n.

The R–T curve at $n=7\frac{2}{3}{n}_0$ and D = −0.21 V nm⁻¹ is fit well (r² = 0.98) by a T² form up to a temperature of 14.7 K and is fit well (r² = 0.996) by a T-linear form.