Magic of high-order van Hove singularity
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Citation Yuan, Noah F. Q. et al. "Magic of high-order van Hove singularity." Nature Communications 10, 1 (December 2019): 5769 © 2019 The Author(s)
As Published http://dx.doi.org/10.1038/s41467-019-13670-9
Publisher Springer Science and Business Media
Version Final published version
Citable link https://hdl.handle.net/1721.1/126294
Terms of Use Creative Commons Attribution 4.0 International license
Detailed Terms https://creativecommons.org/licenses/by/4.0/ ARTICLE https://doi.org/10.1038/s41467-019-13670-9 OPEN Magic of high-order van Hove singularity
Noah F.Q. Yuan 1, Hiroki Isobe1 & Liang Fu1*
The van Hove singularity in density of states generally exists in periodic systems due to the presence of saddle points of energy dispersion in momentum space. We introduce a new type of van Hove singularity in two dimensions, resulting from high-order saddle points and exhibiting power-law divergent density of states. We show that high-order van Hove sin- gularity can be generally achieved by tuning the band structure with a single parameter in moiré superlattices, such as twisted bilayer graphene by tuning twist angle or applying pressure, and trilayer graphene by applying vertical electric field. Correlation effects from high-order van Hove singularity near Fermi level are also discussed.
1 Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. *email: [email protected]
NATURE COMMUNICATIONS | (2019) 10:5769 | https://doi.org/10.1038/s41467-019-13670-9 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-13670-9
he recent discovery of unconventional insulating states and shifts to the Fermi level under gating (the corresponding density superconductivity in twisted bilayer graphene (TBG)1,2 has is within 10% of half filling), the VHS peak in tunneling density of T 45 fi attracted enormous interest. At ambient pressure, these states splits into two new peaks (see also ref. ). These ndings intriguing phenomena of correlated electrons only occur near a clearly demonstrate the prominent role of VHS in magic-angle specific twist angle θ 1:1, widely referred to as the magic TBG. However, it is also known from general consideration and angle. The existence of such a magic angle was first predicted previous STS measurements46–48 that VHS are present at all twist from band structure calculations. Based on a continuum model3,4, angles. It is therefore unclear whether VHS at magic angle is Bistritzer and MacDonald showed5 that the lowest moiré bands in anything special. TBG become exceptionally flat at this magic angle, and are hence In this work, we propose a new perspective that relates magic expected to exhibit strong electron correlation. This pioneering angle to a new type of VHS in the single-particle energy spectrum work inspired a large body of experimental works in recent years, of TBG. We show that as the twist angle decreases below a critical 1,2 θ — which culminated in the discovery reported in refs. . The origin value c, the van Hove saddle point which marks the change of of magic-angle phenomena, i.e., the emergence of super- topology in Fermi surface (Lifshitz transition)—undergoes a conductivity and correlated insulator, is now under intensive topological transition whereby a single saddle point splits into 6–25 θ study . two new ones. Right at c, the saddle point changes from second- While the reduction of bandwidth is undoubtedly important, it order to higher-order; the DOS at VHS is significantly enhanced is evident that the phenomenology of magic-angle TBG cannot be from logarithmic to power-law divergence, which promises ascribed to a completely flat band. Quantum oscillation at low stronger electron correlation. We propose that proximity to such magnetic fields reveals doping-dependent Fermi surfaces within “high-order van Hove singularity”, which requires tuning to the the narrow moiré band. Detailed scanning tunneling spectroscopy critical twist angle or pressure, is an important factor responsible (STS) studies26–28 and compressibility measurements29 show that for correlated electron phenomena in TBG near half filling. the moiré bandwidth at a magic angle is a few tens of meV, larger We demonstrate by topological argument that high-order VHS than what previous calculations found5,30–35. Moreover, there is can be generally achieved by tuning the band structure with just a no direct evidence from existing STS measurements at various single control parameter. For TBG, besides twist angle, pressure fl can also induce superconductivity at a twist angle larger than twist angles that the moiré band is exceptionally at right at the magic angle (see also ref. 28). These experimental results moti- 1:1 49. By tracking the evolution of the moiré band with twist vated us to consider additional feature of electronic structure angle and pressure, we locate the high-order VHS in experi- which may create favorable condition for correlated electron mentally relevant parameter regime and predict its key feature, a phenomena in magic-angle TBG. distinctive asymmetric peak in DOS. This feature compares well One such feature is the van Hove singularity (VHS) in moiré with the experimentally observed VHS peak in magic-angle TBG, bands. The importance of VHS has already been recognized in as shown in Fig. 1. We also discuss the splitting of VHS peak by our theory of correlated TBG36,37 and related studies38–44 using nematic or density wave order. the weak coupling approach. Generally speaking, VHS with divergent density of states (DOS) in two-dimensional systems are Results associated with saddle points of energy dispersion in k space. Ordinary and high-order VHS. In two-dimensional (2D) elec- When a VHS is close to Fermi energy, the increased DOS tron systems with energy dispersion EðkÞ, an ordinary VHS with fi ampli es electron correlation, resulting in various ordering logarithmically diverging DOS occurs at a saddle point ks, instabilities, such as density wave and superconductivity at low determined by temperature36. Indeed, the two recent STS measurements on gate- 26,27 fi tunable TBG around magic angle nd that when the VHS ∇ ¼ ; ð Þ kE 0 and det D<0 1
a 50 b
45 95
40 90
35 85
30 80 × (nS)
25 c
G (nS) 75 20 G+G
15 70
10 = 1.32≈√2 65 5
0 –60 –40 –20 0 20 40 60 1 2 4 6 8 10
E (meV) |E–Ev | (meV)
Fig. 1 Theoretical fit to the tunneling conductance measurement26.aOpen circles are tunneling conductance G of twisted bilayer graphene at twist angle 1:10, and solid lines are fitting according to Eq. (5) with details given in the Supplementary Material Sec. I. Dashed lines denote singularity energies, ¼ : indicating the asymmetry of conductance peaks. b The peak at Ev 16 72 meVp plottedffiffiffi in logarithmic scales. Electron and hole sides of the peak fall into À = η ¼ : ¼ : two parallel lines with the same slope 1 4 and asymmetry ratio 1 32 2. The only parameter in b is the background offset Gc 57 6 nS.
2 NATURE COMMUNICATIONS | (2019) 10:5769 | https://doi.org/10.1038/s41467-019-13670-9 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-13670-9 ARTICLE where D 1 ∂ ∂ E is the 2 ´ 2 Hessian matrix of E at k . Since D these VHS points in the vicinity of this transition is solely ij 2 i j s Λ is symmetric by definition, we can rotate the axes to diagonalize D governed by the local energy dispersion near 0. We now expand EðkÞ near Λ to higher orders: and Taylor expand the energy dispersion near ks as 0 E À E ¼Àαp2 þ βp2, where E is the VHS energy, the À ¼Àα 2 þ β 2 þ γ 2 þ κ 4 þ ¼ ; ð Þ v x y v E Ev p p p p p 3 ¼ À x y x y y momentum p k ks is measured from the saddle point, and Γ the coefficients Àα, β are the two eigenvalues of D with where px (py) is parallel (perpendicular) to M line. In the fi Λ Àαβ ¼ det D<0. This dispersion describes two pieces of Fermi expansion (3), rst-order terms vanish because 0 has by definition zero Fermi velocity. Moreover, only even power terms contours that intersect at the saddle point ks. It is known from a topological consideration that due to the periodicity of EðkÞ on of py are allowed because of the two-fold rotation symmetry of ð ; Þ the Brillouin zone (a torus), van Hove saddle points appear quite TBG, which acts within each valley and maps px py to generally in energy bands of two-dimensional materials50. ð ; À Þ γ κ px py . The third-order term and fourth-order term are We now introduce the concept of high-order VHS associated fi essential to describe the splitting of VHS across the transition, with a high-order saddle point de ned by the following condition: which we shall show below along with the relation to scaling properties. ∇ ¼ ¼ ; ð Þ kE 0 and det D 0 2 The behavior of VHS and Fermi contour of the energy dispersion (3) depends crucially on the sign of β. For β > 0, p ¼ 0 which imply αβ ¼ 0. There exist two types of high-order VHS. (i.e., Λ ) is an ordinary van Hove saddle point with logarithmic fi α ¼ β ¼ ¼ 0 The rst type corresponds to 0, i.e., D 0. The Taylor divergent DOS. The Fermi contour at the VHS energy consistspffiffiffiffiffiffiffiffi of expansion of EðkÞ around such saddle point then starts from at = ¼ α=β two curves that approach straight lines py px ±pffiffiffiffiffiffiffiffi as least the third order, and describes an intersection of three or ! fi φ ¼ ð β=αÞ more Fermi surfaces at a common k point51. Such high-order p 0, intersecting at a nite angle 2 arctan as VHS (type-I) is also referred to as multicritical VHS in the recent shown in Fig. 2d. This behavior corresponds to the VHS in the θ ¼ β ! þ φ ! literature52–54. calculated band structure of TBG at 2 . When 0 , Here we focus instead on a new type of high-order VHS 0 so that the two Fermi contours at the VHS energy tend to touch β ¼ relevant for TBG, where the Hessian matrix D has a single zero tangentially. On the other hand, for < 0, p 0 becomes a local eigenvalue (β ¼ 0), while the other eigenvalue is nonzero (α ≠ 0). energy maximum, while two new saddle points appear at momenta Then in the Taylor expansion of EðkÞ, besides the single second- pffiffiffiffiffiffiffiffiffiffiffiffi order term α we must also include higher order terms in order to Λ ¼ðÀβγ=~γ2; À αβ=~γÞ ð Þ ± ± 2 4 capture the Fermi contours nearby. Importantly, unlike the δ ¼Àαβ2=~γ2 previous case, type-II high-order VHS still describes the touching whosepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi energy is shifted from Ev by < 0, where of two Fermi surfaces, but they touch tangentially (generally ~γ γ2 þ 4ακ. Throughout this manuscript we consider the speaking) rather than intersect at a finite angle as in the case of regime γ2 þ 4ακ > 0 so that ~γ is always real and positive, and two Λ fi Λ Λ ordinary VHS. split saddle points ± are well-de ned. þ and À are a pair of ordinary VHS points related by two-fold rotation, whose Fermi αð Ç β=~γÞ¼ðγ ~γÞ 2 VHS in TBG. We now turn to VHS in TBG, whose low-energy contours are two parabolas 2 px ± py, see Fig. 2f. moiré bands arise from inter-layer coupling of Dirac states on the This behavior corresponds to the VHS in the calculated band two graphene layers. Since single-particle scattering between K θ ¼ : β ! À Λ Λ 0 structure at 1 05 . In the limit 0 , ± approaches 0. and K points requires large momentum transfer, it is suppressed fi Λ Our uni ed description of two regimes of ordinary VHS ( 0 at small twist angles due to the long wavelength of moiré Λ and ± ) using a single local energy dispersion (3) implies that the potential. In the absence of valley hybridization, the moiré bands transition between them corresponds to the sign change of β.In 0 3–5,55 from K and K valleys are decoupled , with energy disper- this process, energy contours at VHS changes from intersecting at ð Þ ð Þ sions denoted by Eþ k and EÀ k , respectively. Time-reversal one point Λ (Fig. 2d and e) to two points Λ (Fig. 2f). Right at ð Þ¼ ðÀ Þ fi 0 ± symmetry implies EÀ k Eþ k , so it suf ces to consider β ¼ 0, p ¼ 0 becomes a high-order VHS point. The Fermi EþðkÞ only in the following. α ¼ðγ ~γÞ 2 contour in its vicinity consists of two parabolas 2 px ± py, The number and location of VHS points in TBG depend on the touching each other tangentially at p ¼ 0. In the special case with twist angle. For example, the band structure calculation for κ ¼ 0, one of the parabolas become a straight line, as shown in θ ¼ 256 reveals the existence of three symmetry-related van Γ Fig. 2e. Hove saddle points on M lines in the mini-Brillouin zone The DOS can be used as an indicator of this topological (MBZ), shown in Fig. 2a. Across the VHS energy, the Fermi transition of VHS. For the energy dispersion (3), we find the DOS contour changes from two disjoint Dirac pockets around MBZ analytically corners to a single pocket enclosing the MBZ center. This leads to Z À 2 the conversion between electron and hole charge carriers, as ρð Þ¼ 1 d p 1 fi 56 E Im 2 þ evidenced by the sign change of Hall coef cient . On the other π ð2πÞ E þ i0 À EðpÞ hand, at θ ¼ 1:05, on each ΓM line there is a local energy "# — — sgn ðαβÞ 1 zþ 2i zÀ maximum instead of saddle point in the moiré valence band. ¼ pffiffiffi Re pffiffiffiffiffi K1À À pffiffiffiffiffi K ΘðÀαβÞ ; As doping increases, an additional hole pocket (not present at 2απ2 zÀ zÀ zþ zþ θ ¼ 2) emerges out of each energy maximum and eventually ð5Þ intersects two Dirac pockets at two new saddle points on opposite qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Γ 32 sides of M . In such band structure there are a total of 6 VHS ¼ðβ=αÞ ðβ=αÞ2 þ ε ε ¼ ~γ2ð À Þ=α3 points, shown in Fig. 2c. where z ± ± and E Ev is By continuity, we deduce from this change in the number of energy deviation from VHS in dimensionless unit. The integra- VHS points that a topological transition of saddle points must tion over p is extended to infinity. Here KðzÞ is the complete occur, at a hitherto unknown critical twist angle, in such a way elliptic integral of the first kind, sgnðrÞ is the sign function Γ Λ fi ð Þ¼À Θð Þ that a saddle point on M (denoted by 0) splits into a pair of de ned as sgn r 1 for r<0 and 1 for r 0, and r Γ Λ 1 ½ À ðÀ Þ Θð Þ¼ new ones off M (denoted by ± ). Importantly, the behavior of 2 1 sgn r is the step function so that 0 0.
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a bcMax y y y k k k
Min k kx kx x
de1 1 f 1
0.5 0.5 0.5 0 0 0 p p p / / / 0 0 y y
y 0 p p p
–0.5 –0.5 –0.5
–1 –1 –1 –0.5 0 0.5 –0.5 0 0.5 –1 –0.5 00.5 1
px/p0 px/p0 px/p0
Fig. 2 Energy contours during the topological transition of van Hove singularity (VHS). Energy contours from band structure calculation are shown in a– c and from the polynomial expansion around saddle point (3) in d–f. Thick lines are energy contours at VHS energy. In band structure calculation, g0 ¼ 1:2 g ¼ ¼ : ¼ – ¼ α=~γ β ¼ : α; κ ¼ β ¼ ; κ ¼ and the parameters are a g 1, b g 1 9, and c g 2. In d f, p0 and the parameters are d 0 2 0, e 0 0, and f β ¼À0:2α; κ ¼À0:2γ2=α.
2 2 a where Λ ¼ 64αβ =~γ is the high-energy cutoff.pffiffiffiffiffiffi In the limit 1 = –0.2 β ! = jβj = 0 0, the prefactor in Eq. (6) diverges as 1 , indicating a = 0.2 strong increase of DOS as the VHS approaches a high-order one. β ¼ fi 0 Right at the transition point 0, we nd power-law divergent /
0.5 DOS near high-order VHS ( À1 C ðE À E Þ 4 E>E ρðEÞ¼pffiffiffiffiffiffiffiffiffi ´ pffiffiffi v v ð7Þ 4 2 À1 0 α~γ ð À Þ 4 –0.1 –0.05 0 0.05 0.1 4 2 Ev E E
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pffiffiffi À = 1 4, and the particle-hole asymmetry ratio is 1.32 close to 2, a b 0.7 which are both consistent with Eq. (7). = 0 α β Besides and , the Fermi contours in momentum space 0.6 = 0.2 depend on both γ and κ, while the DOS in energy domain 2 3 depends only on ~γ. This is because the same scaling dimension of 0.5
2 0 p and p allows the nonlinear transform Q = 2 k χ x y s / γ 1 1 χ 0.4 ~ ¼ À 2; ~ ¼ ; ð Þ px px py py py 9 2α 0.3 À ¼Àα~2þ under which the dispersion (3) becomes E Ev px 3 2 0.2 β~p2 þ ~γ2~p4=ð4αÞ. The nonlinear transform (9) changes the shape y y –2 –1 0 of Fermi contours while it preserves area element and hence 10 10 10 T/E leaves DOS in Eq. (5) invariant. 0 We have thus established by continuity argument the c existence of high-order VHS in the moiré band of TBG at a 0.8 θ θ certain critical twist angle c.Thevalueof c depends on the model we employ. As an illustrative example, we consider the continuum model3–5,32 of TBG, where approximate particle- 0.6 hole symmetry holds, and the Fermi surface topology is solely 0 determined by two dimensionless parameters / 0.4 g ¼ λu=v; g0 ¼ λu0=v.Hereu and u0 denote interlayer hoppings in AA and AB regions respectively, v is the monolayer Dirac velocity and λ is the moiré wavelength. We further fix the ratio 0.2 g0=g ¼ u0=u ¼ 1:2 as calculated for corrugated structure at θ 132.Atgiveng 2½1; 2, we calculate the Taylor coefficients α; β; γ; κ 0 by numerical derivatives of energy with respect to –0.1 –0.05 0 0.05 0.1 momentum at VHS on the hole side, and the result is shown in (E-E )/E Fig. 3b. It can be found that γ2 þ 4ακ is always positive when v 0 2½ ; : β g 1 2 , and across gc 1 995, changes sign (Fig. 3binset) Fig. 4 Particle-hole nesting and density wave instability in twisted bilayer while α; γ; κ do not. With appropriate and reasonable values of graphene. a Six high-order VHS in MBZ, where different colors denote θ Γ u and v,fromgc we can obtain the critical twist angle c 1 , different valleys, and 1, 2, 3 denote three M directions. b Temperature χ χð ; Þ β ¼ : α consistent with experiments. Details of continuum model can dependence of intervalley susceptibility Q Ev at ordinary ( 0 2 ) β ¼ ¼ χ ¼ αÀ1 be found in the Supplementary Material Sec. II. We may and high-order ( 0) VHS with Q 2ks and 0 . c Density of states 31,57,58 ¼ include more ingredients such as lattice relaxation and with (solid) and without (dashed) intervalley density wave. Here E0 strain59 to better describe TBG, thus the Fermi surface topology α3=~γ2 and details can be found in the Methods section. will depend on more details of the model. Nevertheless, the θ existence of high-order VHS and critical twist angle c is 59 robust . we consider two high-order VHS along the same ΓM line but from opposite valleys, χðqÞ is sharply enhanced near wave vector Many-body phenomena near high-order VHS. When chemical q ¼ Q due to the power-law divergent DOS from both valleys, ¼ potential is near high-order VHS, many-body effects can be where Q 2ks is the momentum separation between the two drastic as the strongly diverging DOS leads to divergences in VHS (Fig. 4a). As a result, when approaching low temperatures, noninteracting susceptibilities of various channels. This signals a the intervalley susceptibility χðQÞ of high-order VHS diverges strong tendency to various broken symmetry states around van more rapidly than logarithm (Fig. 4b), indicating a stronger Hove filling, when electron-electron interaction is taken into tendency toward density wave instability than one with ordinary account. Indeed, the recent STS measurements found that when VHS. The DOS peak at VHS energy Ev is split into two in the the Fermi energy approaches the van Hove energy under doping, presence of intervalley density wave (Fig. 4c). the VHS peak in DOS splits into two new ones26,27. A detailed analysis of interacting electrons near high-order Discussion VHS is beyond the scope of this work. Instead we discuss two To summarize, we propose that the proximity to high-order possible scenarios for the observed splitting of the VHS peak near VHS underlies correlated electron phenomena in TBG near the Fermi energy. magic angle. We reveal a distinctive feature of high-order VHS First, strains in experimental samples can split DOS peak by —power-law divergent DOS with an asymmetric peak, breaking rotation symmetries which relate VHS along different which compares well with the recent STS data on magic-angle directions. Though the energy splitting due to strain may be small TBG. We also discuss nematic and density wave instabilities at single-particle level, Coulomb interaction U can give rise to the due to electron-electron interaction near van Hove filling, ð À χ ÞÀ1 Stoner-type enhancement factor 1 U s within the random and illustrate the splitting of VHS in these broken symmetry phase approximation (RPA). Since the noninteracting suscept- states. χ fl ibility s re ects the divergent DOS, strain effect on high-order It is worth examining the similarity and difference between the VHS can be greatly enhanced by interaction. two scenarios favoring correlated electron phenomena in TBG: Second, an intervalley density wave order can split a DOS peak flat band5 and (high-order) VHS. Both scenarios rely on large by spontaneously breaking translation symmetry. To show such a DOS enhancing electron correlation. Obviously, the largest pos- density wave instability, we calculate intervalley susceptibility sible DOS is realized in the limit of a completely flat band60. χðqÞ with finite momentum q. There are in total six high-order However, a completely flat band is difficult to achieve under VHS from two valleys, located at three ΓM lines (Fig. 4a). When realistic conditions. On the other hand, the VHS scenario is more
NATURE COMMUNICATIONS | (2019) 10:5769 | https://doi.org/10.1038/s41467-019-13670-9 | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-13670-9 likely to be relevant when the bandwidth is not so small in Received: 8 July 2019; Accepted: 14 November 2019; comparison to electron interaction. In this scenario, the DOS near Fermi energy matters more than those far away, and it is further increased by proximity to high-order VHS. Importantly, to achieve high-order VHS generally requires tuning the band structure with only a single parameter, whereas to achieve a References completely or extremely flat band usually requires more fine 1. Cao, Y. et al. Unconventional superconductivity in magic-angle graphene tuning. superlattices. Nature 556, 43 (2018). Broadly speaking, moiré superlattices in 2D materials offer an 2. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle unprecedented platform for studying correlated electron phe- graphene superlattices. Nature 556, 80 (2018). 3. Lopes dos Santos, J. M. B., Peres, N. M. R. & Castro Neto, A. H. Graphene nomena near VHS. Electrostatic gating enables doping these Bilayer with a twist: electronic structure. Phys. Rev. Lett. 99, 256802 (2007). systems to van Hove filling without introducing disorder. The 4. Lopes dos Santos, J. M. B., Peres, N. M. R. & Castro Neto, A. H. Continuum great tunability of moiré band structure by twist angle, electric model of the twisted graphene bilayer. Phys. Rev. B 86, 155449 (2012). field, strain and other means grants access to high-order VHS, 5. Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer where the strongly divergent DOS promises a plethora of many- graphene. Proc. Natl Acad. Sci. 108, 12233 (2011). 6. Xu, C. & Balents, L. Topological Superconductivity in twisted multilayer body phenomena. Such examples include TBG discussed in the graphene. Phys. Rev. Lett. 121, 087001 (2018). maintext and trilayer graphene on boron nitride discussed in 7. Volovik, G. E. Graphite, graphene and the flat band superconductivity. Pis’ma Supplementary Material Sec. III. The time is coming for creating v. ZhETF 107, 537 (2018). [JETP Letters 107, 516 (2018)]. strong electron correlation by designing single-particle dispersion 8. Yuan, N. F. Q. & Fu, L. Model for the metal-insulator transition in graphene around VHS. superlattices and beyond. Phys. Rev. B 98, 045103 (2018). 9. Po, H. C., Zou, L., Vishwanath, A. & Senthil, T. Origin of Mott insulating behavior and superconductivity in twisted bilayer graphene. Phys. Rev. X 8, Methods 031089 (2018). Fitting of tunneling conductance peaks. In Eq. (5) we assume the dispersion (3) 10. Dodaro, J. F., Kivelson, S. A., Schattner, Y., Sun, X. Q. & Wang, C. Phases of a extends to the whole momentum space, which holds only for a finite momentum phenomenological model of twisted bilayer graphene. Phys. Rev. B 98, 075154 range in realistic systems. Assuming the dispersion over the whole momentum (2018). E fi j j Λ space as p, whose Taylor expansion is Ep in the nite momentum range p < , 11. Padhi, B., Setty, C. & Phillips, P. W. Doped twisted bilayer graphene near we find the total DOS of such dispersion can be written as magic angles: proximity to wigner crystallization not mott insulation. Nano Z 2 Lett. 10, 6175 (2018). ρ ð Þ¼ d p δð ÀE Þ¼ρð Þþρ ; ð Þ 12. Xu, X. Y., Law, K. T. & Lee, P. A. Kekulé valence bond order in an extended tot E 2 E p E c 10 ð2πÞ Hubbard model on the honeycomb lattice, with possible applications to where ρðEÞ is Eq. (5) if the Taylor expansion is Eq. (5), and ρ is the background twisted bilayer graphene. Phys. Rev. B 98, 121406 (2018). c fi contribution 13. Kang, J. & Vafek, O. Strong coupling phases of partially lled twisted bilayer Z graphene narrow bands. Phys. Rev. Lett. 122, 246401 (2019). 2 ρ ¼ d p ½δð ÀE ÞÀδð À Þ: ð Þ 14. Thomson, A., Chatterjee, S., Sachdev, S. & Scheurer, M. S. Triangular c 2 E p E Ep 11 jpj>Λ ð2πÞ antiferromagnetism on the honeycomb lattice of twisted bilayer graphene. When DOS of the tip is featureless, the tunneling conductance from the tip to Phys. Rev. B 98, 075109 (2018). ρ 15. Lian, B., Wang, Z. & Andrei Bernevig, B. Twisted Bilayer graphene: a phonon the sample will be proportional to the total DOS tot of the sample. There are two distinct types of divergent DOS and hence conductance peaks in this work and also driven superconductor. Phys. Rev. Lett. 122, 257002 (2019). in experimental data. The first type is due to ordinary VHS, which is symmetric 16. Venderbos, J. W. F. & Fernandes, R. M. Correlations and electronic order in a and logarithmic, and another type is due to high-order VHS, which is intrinsically two-orbital honeycomb lattice model for twisted bilayer graphene. Phys. Rev. asymmetric and power-law. These two types of conductance peaks follow different B 98, 245103 (2018). functional forms as follows (o denotes ordinary VHS and h denotes high-order 17. Fidrysiak, M., Zegrodnik, M. & Spalek, J. Unconventional topological VHS) superconductivity and phase diagram for an effective two-orbital model as applied to twisted bilayer graphene. Phys. Rev. B 98, 085436 (2018). ð Þ¼ ´ j À jÀ ; ð Þ Go E A log E Ev Gc 12 18. Su, Y. & Lin, S.-Z. Pairing symmetry and spontaneous vortex-antivortex lattice
À1 in superconducting twisted-bilayer graphene: Bogoliubov-de gennes approach. ð Þ¼ ´ ð À Þ 4 À ; ð Þ Gh E A Re Ev E Gc 13 Phys. Rev. B 98, 195101 (2018). each with three parameters A; E and G . Among them, E denotes the energy of 19. Guinea, F. & Walet, N. R. Electrostatic effects, band distortions, and v c v superconductivity in twisted graphene bilayers. Proc. Natl Acad. Sci. USA 115, conductance peak, and Gc is the background contribution. In Fig. 1b, we plot conductance data G near the peaks at twist angle 1:10, 13174 (2018). where energy and conductance are both plot in logarithmic scale. Since the VHS is 20. Ochi, M., Koshino, M. & Kuroki, K. Possible correlated insulating states in high-order, conductance peak is asymmetric and power-law, two sides of the peak magic-angle twisted bilayer graphene under strongly competing interactions. will follow two parallel lines with the same slope −1/4 but different vertical Phys. Rev. B 98, 081102 (2018). intercepts. 21. Wu, X.-C., Jian, C.-M. & Xu, C. Coupled wire description of the correlated We make the substitution E ! E þ iη in the expressions above of tunneling physics in twisted bilayer graphene. Phys. Rev. B 99, 161405 (2019). conductance to describe broadening effect. The two sides of the DOS peak of 22. Wu, F., MacDonald, A. H. & Martin, I. Theory of phonon-mediated ordinary VHS are broadened in the same way, while the two sides of the DOS peak superconductivity in twisted bilayer graphene. Phys. Rev. Lett. 121, 257001 of high-order VHS are broadened differently due to the intrinsic particle-hole (2018). asymmetry. As a result, after broadening, Ev coincides with the DOS peak energy 23. Wu, F. Topological chiral superconductivity with spontaneous vortices and Em when VHS is ordinary, while for high-order VHS, Ev can deviate from Em as supercurrent in twisted bilayer graphene. Phys. Rev. B 99, 195114 (2019). shown in Fig. 1a. 24. Ray, S. & Das, T. Wannier pairs in the superconducting twisted bilayer graphene and related systems. Phys. Rev. B 99, 134515 (2019). Mean-field Hamiltonian of interacting high-order VHS. With finite intervalley 25. Zhu, G.-Y., Xiang, T. & Zhang, G.-M. Inter-valley spiral order in the Mott density wave order parameter Δ, the low-energy Hamiltonian of two high-order insulating state of a heterostructure of trilayer graphene-boron nitride. Sci. VHS on the same ΓM line can be written as HðpÞ¼E À αp2 þ κp4 þ γp p2τ þ Bull. 63, 1087 (2018). v x y x y z 26. Kerelsky, A. et al. Maximized electron interactions at the magic angle in Δτ ; where Pauli matrices τ act on the valley indices. From this Hamiltonian x twisted bilayer graphene. Nature 572,95–100 (2019). Fig. 4c is plotted, with order parameter Δ ¼ 0:01E and broadening 0 27. Choi, Y. et al. Electronic correlations in twisted bilayer graphene near the η ¼ 5 ´ 10À3E . 0 magic angle. Nat. Phys. https://doi.org/10.1038/s41567-019-0606-5 (2019). 28. LeRoy, B. Topologically protected helical states in minimally twisted bilayer Data availability graphene. Phys. Rev. Lett. 121, 037702 (2018). The data that support the findings of this study are available from the corresponding 29. Tomarken, S. L. et al. Electronic compressibility of magic angle graphene author upon reasonable request. superlattices. Phys. Rev. Lett. 123, 046601 (2019).
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Nematic superconductivity We thank Alex Kerelsky and Abhay Narayan Pasupathy for sharing experimental data stabilized by density wave fluctuations: application to twisted bilayer with us, and Ali Yazdani for discussions. This work is supported by DOE Office of Basic graphene. Phys. Rev. B 99, 144507 (2019). Energy Sciences, Division of Materials Sciences and Engineering under Award DE- 38. Nandkishore, R., Levitov, L. S. & Chubukov, A. V. Chiral superconductivity SC0018945. L.F. is partly supported by the David and Lucile Packard Foundation. from repulsive interactions in doped graphene. Nat. Phys. 8, 158 (2012). 39. Laksono, E. et al. Singlet superconductivity enhanced by charge order in Author contributions nested twisted bilayer graphene Fermi surfaces. Solid State Commun. 282,38 L.F. conceived the idea and designed the work. N.F.Q.Y. performed band structure and (2018). DOS calculations. N.F.Q.Y., H.I., and L.F. contributed to theoretical analysis. N.F.Q.Y. 40. Sherkunov, Y. & Betouras, J. J. Electronic phases in twisted bilayer graphene at and L.F. wrote the manuscript. magic angles as a result of Van Hove singularities and interactions. Phys. Rev. B 98, 205151 (2018). 41. Kennes, D. M., Lischner, J. & Karrasch, C. Strong correlations and d þ id Competing interests superconductivity in twisted bilayer graphene. Phys. Rev. B 98, 241407 (2018). The authors declare no competing interests. (R). 42. Liu, C.-C., Zhang, L.-D., Chen, W.-Q. & Yang, F. Chiral spin density wave and Additional information d þ id superconductivity in the magic-angle-twisted bilayer graphene. Phys. Supplementary information Rev. Lett. 121, 217001 (2018). is available for this paper at https://doi.org/10.1038/s41467- 019-13670-9. 43. Lin, Y.-P. & Nandkishore, R. M. Kohn-Luttinger superconductivity on two orbital honeycomb lattice. Phys. Rev. B 98, 214521 (2018). Correspondence 44. González, J. & Stauber, T. Kohn-Luttinger superconductivity in twisted bilayer and requests for materials should be addressed to L.F. graphene. Phys. Rev. Lett. 122, 026801 (2019). Peer review information Nature Communications thanks the anonymous reviewer(s) for 45. Liu, Y.-W. et al. Magnetism near half-filling of a van Hove singularity in their contribution to the peer review of this work. twisted graphene bilayer. Phys. Rev. B 99, 201408 (2019). 46. Li, G. et al. Observation of Van Hove singularities in twisted graphene layers. Reprints and permission information Nat. Phys. 6, 109 (2010). is available at http://www.nature.com/reprints 47. Wong, D. et al. Local spectroscopy of moiré-induced electronic structure in Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in gate-tunable twisted bilayer graphene. Phys. Rev. B 92, 155409 (2015). published maps and institutional affiliations. 48. Kim, K. et al. Tunable Moiré bands and strong correlations in small-twist- angle bilayer graphene. Proc. Natl Acad. Sci. USA 114, 3364 (2017). 49. Yankowitz, M., et al., Tuning superconductivity in twisted bilayer graphene, Science https://doi.org/10.1126/science.aav1910 (2019). Open Access This article is licensed under a Creative Commons 50. van Hove, L. The occurrence of singularities in the elastic frequency Attribution 4.0 International License, which permits use, sharing, distribution of a crystal. Phys. Rev. 89, 1189 (1953). adaptation, distribution and reproduction in any medium or format, as long as you give 51. Biswas, R. R., Fu, L., Laumann, C. R. & Sachdev, S. SU(2)-invariant spin appropriate credit to the original author(s) and the source, provide a link to the Creative liquids on the triangular lattice with spinful Majorana excitations. Phys. Rev. B Commons license, and indicate if changes were made. The images or other third party 83, 245131 (2011). material in this article are included in the article’s Creative Commons license, unless 52. Shtyk, A., Goldstein, G. & Chamon, C. Electrons at the monkey saddle: a indicated otherwise in a credit line to the material. If material is not included in the multicritical lifshitz point. Phys. Rev. B 95, 035137 (2017). article’s Creative Commons license and your intended use is not permitted by statutory 53. Efremov, D. V.et al. Multicritical Fermi surface topological transitions, regulation or exceeds the permitted use, you will need to obtain permission directly from arXiv:1810.13392 [cond-mat.str-el] (2018). the copyright holder. To view a copy of this license, visit http://creativecommons.org/ 54. Tamai, A. et al. Fermi surface and van Hove singularities in the itinerant licenses/by/4.0/. metamagnet Sr3Ru2O7. Phys. Rev. Lett. 101, 026407 (2008). 55. Cao, Y. et al. Superlattice-induced insulating states and valley-protected orbits in twisted bilayer graphene. Phys. Rev. Lett. 117, 116804 (2016). © The Author(s) 2019
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