Isospin competitions and valley polarized correlated insulators in twisted double bilayer graphene

New phase of matter usually emerges when a given symmetry breaks spontaneously, which can involve charge, spin, and valley degree of freedoms. Here, we report an observation of new correlated insulators evolved from spin-polarized states to valley-polarized states in twisted double bilayer graphene (TDBG) driven by the displacement field (D). At a high field |D | > 0.7 V/nm, we observe valley polarized correlated insulators with a big Zeeman g factor of ~10, both at v = 2 in the moiré conduction band and more surprisingly at v = −2 in the moiré valence band. Moreover, we observe a valley polarized Chern insulator with C = 2 emanating at v = 2 in the electron side and a valley polarized Fermi surface around v = −2 in the hole side. Our results demonstrate a feasible way to realize isospin control and to obtain new phases of matter in TDBG by the displacement field, and might benefit other twisted or non-twisted multilayer systems.

T wisted graphene-based moiré system is an ideal platform to discover new quantum phases of matter [1][2][3][4][5][6][7][8][9][10][11][12][13][14] . The formation of narrow moiré flat bands quenches the kinetic energy 15 , and induces correlated phases as a result of an enhanced Coulomb interaction with respect to the kinetic energy. Each conduction or valence moiré band has four isospin flavors, involving spin and valley degree of freedoms. The interplay between charge, spin and valley (orbital) yields a rich phase diagram, in which some delicate yet fragile quantum phases are accessible at extreme conditions. The pronounced electron interaction in flat bands tend to induce a spontaneous symmetry breaking at integer fillings [16][17][18] . For instance, valley polarization helps to unveil the nontrivial topological nature of the moiré bands and quantized anomalous Hall effect at odd fillings in hexagonal boron nitride (h-BN) aligned twisted bilayer graphene (TBG) 19,20 and ABC trilayer graphene 9 as well as twisted monolayer-bilayer graphene 11,12 , while spin polarization is critical to the formation of correlated insulator at v = 2 in TDBG [5][6][7][8]21,22 . Intuitively, one might ask if it is possible to drive the polarized states from one flavor to the other, and vice versa. This interesting question has been partially targeted in recent observations of the Pomeranchuk effect in TBG which involves a transition of isospin from unpolarized to polarized states 23,24 . However, it remains open whether one could achieve a tunable transition of polarized states from one isospin to the other, and moreover whether this transition will lead to new phase of matter.
Among many twisted graphene-based moiré systems, TDBG is a promising target to address this question based on the following concerns. First, TDBG is a strongly displacement field (D) tunable moiré system 25,26 . The displacement field can in-situ tune the relative strength of Coulomb interactions, relative to its kinetic energy, thus acts an extra knob beyond the twist angle to tune the band structure and topological properties of TDBG. Second, correlated insulator at half filling in TDBG are spin polarized 5-7 , and it emerges over a wide range of twist angle from~1.1°tõ 1.5°, with a finite D varying from~0.2 to~0.5 V/nm. The symmetry breaking instability occurs at the boundary between spin-polarized and unpolarized states in TDBG, and it give rise to quantum critical behaviors 22 . Besides, TDBG might, in principle, host valley-polarized states as the ground states since the perpendicular magnetic field could induce a competition between spin and valley by spin and orbital Zeeman effects 27 , thus resulting to an isospin transition from one to the other.
In this work, we report a study of isospin polarizations in TDBG. We fabricate ultra-clean TDBG devices with gold top gate and graphite bottom gate. We push the displacement field to its limit, and importantly we unveil a transition from spin polarization to valley polarization when D approaches a critical field D*. The transition is accompanied by new phases of matter, i.e. a valley polarized Chern insulator state at v = 2 in the electron side and a valley polarized Fermi surface around v = −2 in the hole side, which never realized in previous twisted graphene-based moiré system. The valley Chern insulator shows a well quantized Hall conductance plateau at 2e 2 /h and correspondingly a vanishing longitudinal component. The valley polarized Fermi surface shows a series of quantized Landau levels (LLs) with v LL = 0, ±1, ±2, ±3, ±4 and others in the landau fan diagram.

Results and discussion
General information of TDBG devices. The TDBG devices are fabricated by the cut and stack technique 28 . A big flake of ABstacked bilayer graphene is firstly cut into two pieces then stacked up with a twist angle of 60°+ θ, so called AB-BA TDBG, as shown in Fig. 1a, where θ is~1.3°to ensure a strong electron correlation of TDBG 5 . Compared to the previously studied TDBG with a twist angle of θ (AB-AB TDBG), AB-BA TDBG has similar band structure yet different topological properties according to the theoretical calculations 26,[29][30][31][32] . The stacked samples are ultraclean with a bubble-free area over a length scale of~20 μm, as shown in Fig. 1a and Supplementary Fig. 5a. These devices have a dual-gate configuration with a graphite bottom gate and gold top gate, which allows independent tuning of the carrier density n and D. Here, n ¼ ðC BG V BG þ C TG V TG Þ=e and D ¼ ðC BG V BG À C TG V TG Þ=2ε 0 ; where C BG (C TG ) is the geometrical capacitance per area for bottom (top) gate, e is the electron charge, and ε 0 is the vacuum permittivity. These devices show a good quality with the angle inhomogeneity <0.01°within 2 μm size and mobility on order of 10 5 cm 2 /(V·s) in Fig. 1b, c and Supplementary Fig. 5.
We perform cryogenic magneto-transport measurements at a base temperature of T = 1.8 K. Figure 1d shows the longitudinal resistance (R xx ) as a function of filling factor (v) and D at B = 0 T. The filling factor is defined as v = 4n/n s , corresponding to the number of carriers per moiré unit cell. Here, n s = 4/A ≈ 8θ/(√3a 2 ), where A is the area of a moiré unit cell, θ is twisted angle, and a is the lattice constant of graphene. In Fig. 1d, two evident resistance peaks at v = −4 and 4 correspond to the moiré band gap, and the resistance peak at v = 0 indicates a band gap opening between the conduction band (CB) and the valance band (VB) which results from D induced inversion symmetry breaking 33 . Correlated insulators at half filling v = 2 are observed at a medium D from 0.3 to 0.6 V/nm in device D1 (Fig. 1d) and a similar D from 0.2 to 0.5 V/nm in device D2 (Fig. 2a). These correlated insulators are spin polarized as evidenced from its positive in-plane magnetic field response (Fig. 2b, d) and a corresponding Zeeman spin g factor g s of~2.25 (Fig. 2f). Similarly, a spin-polarized insulator is developed at v = 1 in Fig. 2b and d, with g s of~2.35 (Fig. 2f). All these features both at zero magnetic field and in-plane magnetic field are in agreements with previous results of AB-AB TDBG [5][6][7] , which are also in line with the identical band structure between AB-BA and AB-AB TDBG 26 .
Valley polarized correlated insulators at high displacement fields. We observe new correlated insulators in the hole side at v = −2 when D is sufficiently large. The new correlated insulators are manifested as new resistance peaks in the hole side at v = −2, as shown in Fig. 1e when |D | > 0.75 V/nm for device D1 and in Fig. 2c when |D | > 0.6 V/nm for device D2, in the presence of a finite perpendicular magnetic field. It is unexpected as the valance band becomes more and more dispersive and entangles with remote bands with the increase of |D | , and it demands an important role played by electron correlation. In principle, valley degeneracy can be lifted in a perpendicular magnetic field 34,35 , as shown in Fig. 1f. Considering a Bloch electron is subjected to a weak magnetic field, the energy can be expressed as ε N;σ;τ k; where the second (third) term is the spin (orbital) Zeeman energy. Here, N is the band index, σ = ±1 is the spin index, S = 1/2 is spin quantum number, and τ = ±1 is the valley index. In the third term, m n;τ k ð Þ corresponds to the orbital magnetization, and it is opposite in different valley K and K' according to the time reversal symmetry. Therefore, the valley degeneracy can be lifted with the orbital Zeeman effect. The last two terms can be expressed as gμ B B, where g is the effective g factor (g = S*g s = 1 for pure spin Zeeman effect).
We perform Zeeman effect measurements for the new correlated insulator at v = −2. The resulting Zeeman thermal gap as a function of B ⊥ is plotted in Fig. 1g, and from which we obtain effective Zeeman g ≈ 13.6. This insulator is also not related to spin degree of freedom, as it only response to perpendicular magnetic field and it is featureless even at B || = 9 T (Fig. 2b).
Thus, we assign the correlated insulator at v = −2 as a valleypolarized insulator.
Moreover, we also observe a new correlated resistance peak at electron side v = 2 when |D | > 0.9 V/nm, aside from the conventional spin correlated insulator at v = 2 when D is at mediate range from 0.3 to 0.6 V/nm, as shown in Fig. 1e. At a higher B ⊥ , the insulator at the higher D is much more pronounced (Fig. 1e), while the spin-polarized insulator at the lower D becomes weaker and weaker and eventually disappear at B ⊥ > 5 T (Supplementary Fig. 13). We plot the gap for the new insulator at higher D as a function of B ⊥ in Fig.1g, and once again we get g ≈ 8.5, indicating a dominating role played by valley instead of spin. Thus, the correlated insulator at v = 2 when | D | > 0.9 V/nm is also valley polarized, similar to that at v = −2.
Competition of isospin polarization between spin and valley. The disappear of spin-polarized insulator and simultaneously the rise of the valley-polarized insulator at v = 2 with increase B ⊥ suggest a competition between spin and valley polarizations 35,36 . Such a competing scenario is even more pronounced in the valence band at v = −2. The evolution of R xx at v = −2 with B || is studied at a fixed B ⊥ = 6 T in D2, as shown in Fig. 2e. Note that the orbital Zeeman splitting energy remains almost unchanged and only the spin Zeeman effect need to be considered in this situation. The peak resistance at v = −2 decreases as B || increases, and it shows an unconventional insulating behavior where R(T) doesn't present a well thermal activation behavior. Instead, it could be divided into two parts, i.e. a strong B || dependent and T sensitive insulating behavior at T < 10 K and an almost B || independent insulating state at T > 16 K. According to the thermal activation behavior R xx ¼ R 0 exp 4=ð2k B TÞ À Á , where k B is the Boltzmann constant, we estimated the thermal energy gaps (4) from R (T) at T < 10 K in Fig. 2g. We can see that 4 decreases as B || increases, and the spin g factor g s ≈ −1.26. The negative sign of g s indicates that spin Zeeman effect tend to close the gap of the valley polarized insulator at v = −2, a strong evidence of competing instead of cooperating between spin and valley polarization. Similar competing behaviors and negative g s are also observed at v = 2 in device D1 (Supplementary Fig. 7).
Valley Chern insulator with C = 2 emanating from v = 2. The topological nature of the valley polarized moiré flat band can be revealed in the Landau fan diagram 18,[37][38][39][40][41][42] . Figure 3a shows the longitudinal resistance (R xx ) as a function of v and B ⊥ at D = 0.8 V/nm and T = 1.8 K. An obvious wedge-shaped R xx minimum emanating from v = 2 develops along the line with a slope of dn/dB = Ce/h, where C is equal to 2. The vanishing R xx comes together with a plateau of Hall resistance (R xy ), with an onset magnetic field of B ⊥ = 4.2 T. The perfect quantization of R xy as well as corresponding zero R xx , i.e. σ xy = 2e 2 /h with σ xx0 , is demonstrated in Fig. 3b at a fixed B ⊥ = 7.3 T and Figure 3c at a fixed carrier density of v = 2.46. Here, the conductance σ xx and σ xy are given by:   σ xy ¼ R xy =ðða r Ã R xx Þ 2 þ R xy 2 Þ, where a r is the aspect ratio of the device. Similar quantization with C = 2 is also observed in other three devices with both AB-BA and AB-AB stacking, as shown in the Supplementary Fig. 9-11. The observed quantization with C = 2 is in good agreement with our theoretic calculation of the valley polarized moire Chern band in TDBG (see Supplementary Note 1 for details), where C v = 1 and the total Chern number C = 2*C v by including two-fold spin degeneracy.
The magnetic field dependent 4 for the C = 2 valley Chern insulator is illustrated in the inset of Fig. 3c. The gap 4 increases with B ⊥ and tends to saturate at high B ⊥ , being consistent with the mechanism that the valley polarization becomes stronger with the increased B ⊥ . We compare the gap of valley Chern insulator with that of the first LL (v LL = +6) emanating from v = 0 at B ⊥ = 6 T ( Supplementary Fig. 8), and it turns out that the energy gap of the former (~4.3 meV) is much larger than that of the latter (~1.6 meV). The valley Chern insulator is also clearly distinguished from the faint features of LLs (with v LL = +1, +3, +4) emanating from v = 2 in the fan diagram of Fig. 3a, where the R xx dip with C = 2 reaches zero while the rest not. The distinguishment is further illustrated in the perfect quantization of Hall conductance with C = 2 in Fig. 3b, c.
Fermi surface reconstruction and Landau quantization around v = −2. In contrast to the topological valley subband in CB, it is a trivial valley polarized subband with C = 0 in VB according to our calculations with Hartree-Fock approximation at a large D, and the VB is generally more dispersive than CB as D is increased (see Supplementary Note 1 for details). It goes through a series of transformations due to enhanced valley polarization as B ⊥ is increased. Figure 4 illustrates typical Fermi surface reconstructions driven by valley polarization at different magnetic field when a large D = −0.73 V/nm is applied. Essentially, the reconstruction is captured in the change of Hall filling factors v H = 4n H /n to moiré band filling factor v, where Hall carrier density n H = B ⊥ /eR xy .
At B ⊥ = 0.8 T, the magnetic field is too small to reconstruct the energy band, and the v H in Fig. 4c varies linearly, indicated by two blue dashed lines of v H = v and v H = v + 4 near the charge neutrality point (CNP) and moiré band edge, respectively. van Hove singularity (VHS) is indicated by the diverging Hall carrier density near v = −2, which suggests a Lifshitz transition of the Fermi surface 39,43 . Note that VHS locates at around v = −2, corresponding to the flat band of the highest VB, based on our band structure calculations from the continuum model (Supplementary information). The position of VHS changes with the increased electric field, and it follows the white contour line in Fig. 4b. At B ⊥ = 4 T, the Fermi surface of VB is greatly reconstructed (Fig. 4e-g). The divergent carrier densities are reset to zero at v = −2 and extend to both sides along the line of v H = v + 2. In this case, a correlation driven energy gap replaces the VHS and separates the VB into two valley-polarized subbands with new VHSs near v = −1 and v = −3.
The reconstruction of VB is reminiscent of the stoner criterion in ferromagnetic metal 44 . The paramagnetic phase becomes of instability if U*g(E F ) > 1, where U and g(E F ) are the Coulomb repulsion and density of states (DOS) at the Fermi surface, respectively. Hence, similar to the stoner mechanism, the occupancy of VB is redistributed between two valley flavors facilitated by the divergent DOS at VHS, and then the orbital Zeeman effect separate the two valley-polarized subbands with the increased B ⊥ .
The complete Landau fan diagram is present in Fig. 4i.  The even LLs are derived from the valley-polarized subbands, while the odd LLs correspond to the quantum hall ferromagnets 45,46 which polarize the spin flavors. The difference is reflected in the energy scale between them. As shown in Fig. 4k, even LLs with v LL = ±2, −4 reveal well quantized Hall plateaus and zero R xx . However, odd LLs with v LL = ±1, −3, −5 show finite R xx and incipient Hall plateau which indicates incomplete quantization. The perpendicular magnetic field plays a key role on the reconstruction of VB. It not only interacts with opposite orbital magnetic moments at different valleys, promoting the formation of valley-polarized states, but also makes the carriers do the cyclotron motion in valleypolarized subbands.
We have observed new correlated insulators evolved from spinpolarized states to valley-polarized states in TDBG. The transition of the isospin polarization is a result of the competition between spin and valley, driven by the displacement field and magnetic field. Moreover, we have unveiled the unique topology of the TDBG, including a quantized valley Chern insulator with C = 2 emanating at v = 2 in the electron side and a valley polarized yet topologically trivial Fermi surface with C = 0 around v = −2 in the hole side. It is worth mentioning that our devices are of high quality and state-of-the-art clean, with twist angle inhomogeneity <0.01°. Our studies shed light on the importance of the displacement field to tune the band topology, and our results could enrich the current understanding of the TDBG system and provide references for other twisted or non-twisted multilayer systems as well.

Methods
Estimation of the twisted angle. The top and bottom gate capacitance can be extracted from the Landau fan diagram and dual-gate voltage mapping. We calculate the carrier density n and the electric displacement field D according to the capacitances and transform the R xx (V b , V t ) mapping to the R xx (n, D) mapping. According to the formula, n s = 4/A ≈ 8θ/(√3a 2 ), the twisted angle θ can be extracted from the location of superlattice resistance peak in the R xx (n, D) mapping. The obtained twist angle is further corrected by the Brown-Zak oscillations in the Landau fan diagram. As shown in Supplementary Fig. 10, R xx shows dips at Φ/Φ 0 = 1/q, where Φ = AB ⊥ is the magnetic flux per moiré unit cell, Φ 0 = h/e is the magnetic flux quantum, and q is a positive integer.
Estimation of the Zeeman g factor. The Zeeman splitting energy, including the contributions from both spin and orbital magnetization, can be expressed as 4~2gμ B B, where g corresponds the effective g factor. The value of g can be extracted from the linear fitting of thermal activation energy gap versus magnetic field. Under the in-plane magnetic field, only the spin Zeeman effect needs to be considered, hence 4~2gμ B B = 2Sg s μ B B, where S = 1/2 is the spin quantum number and g s is spin g factor. To clearly show the role of spin, we present our results using g s instead of g in this situation.
Calibration of the magnetic field direction. The change of the magnetic field direction is achieved by rotating the sample with the attocube stage. Then the direction is further calibrated by measuring Hall resistance at different rotating angle. We measured the Hall resistance at V tg = V bg = 0 V from −3 T to 3 T in perpendicular and parallel direction, as shown in Supplementary Fig. 15. The response of Hall resistance totally come from the cyclotron motion of the electron under the perpendicular magnetic field. Hence, the value of the Hall resistance under the parallel magnetic field can be used to calibrate the direction of the magnetic field. The R xy is about −30 ohm at B || = 3 T, which is equivalent to the R xy at B ⊥ = 0.03 T. To better visualize the difference, we shrink X-axis of the parallel magnetic field by~100 times, i.e. −30 mT to 30 mT, while keeping the Xaxis of the perpendicular direction unchanged; the almost coincident lines suggest the residual perpendicular component is~30 mT at B || = 3 T, and thus the error of parallel magnetic field is about 1%. Since it needs a large perpendicular magnetic field to realize the valley-polarized states, thus the small error has no effect on our experiment results.