The nature of correlations in the insulating states of twisted bilayer graphene
TThe nature of correlations in the insulating states of twisted bilayer graphene
J.M. Pizarro, M.J. Calder´on, and E. Bascones ∗ Materials Science Factory. Instituto de Ciencia de Materiales de Madrid(ICMM). Consejo Superior de Investigaciones Cient´ıficas (CSIC),Sor Juana In´es de la Cruz 3, 28049 Madrid (Spain). (Dated: January 6, 2020)The recently observed superconductivity in twisted bilayer graphene emerges from insulatingstates believed to arise from electronic correlations. While there have been many proposals toexplain the insulating behaviour, the commensurability at which these states appear suggests thatthey are Mott insulators. Here we focus on the insulating states with ± I. INTRODUCTION
Unexpected insulating states have been recently ob-served upon doping a graphene bilayer with a small twistangle ∼ . o − . o [1]. The stacking misorientationcreates a moir´e pattern with a superlattice modulationcorresponding to thousands of atoms per unit cell. Theinsulating states, believed to be due to electronic cor-relations [1], were originally observed when the chargeper moir´e cell is ± ± ∗ Electronic address: [email protected] erties of the density of states at the Fermi level [8–15].However, an important clue comes from the fact that thefillings at which these states are experimentally foundcorrespond to an integer number of electrons/holes permoir´e unit cell, as for Mott states in Hubbard models. Infact, the exact commensurability has been now observedfor a larger range of twist angles [1–3], for which con-siderably different density of states are expected. Theproximity of a metallic gate, at a distance of only 1 − U c which de-pends on the filling and on other interaction parame-ters such as Hund’s coupling J H . Though localized elec-trons have a tendency to order magnetically, the Motttransition does not require any symmetry breaking. Forinteractions smaller than the critical interaction for theMott transition, a metallic correlated state is found withthe spectral weight partially redistributed to many-bodyHubbard bands.Local correlations, namely correlations between elec-trons on the same site, induce a Mott transition at integerfillings via the opening of a charge gap. Approaches usu-ally used to address the Mott transition, such as single- a r X i v : . [ c ond - m a t . s t r- e l ] J a n site Dynamical Mean Field Theory (DMFT) [17] or slaveparticle techniques [18–21], only include these on-site cor-relations. However, even if in the Hubbard model in-teractions are restricted to electrons on the same site,inter-site (non-local) correlations are established. Theinterplay between these magnetic and orbital non-localcorrelations and the charge degree of freedom affects sig-nificantly the Mott transition. Here we show that exper-imental observations on the insulating states in TBG arenot consistent with expectations including only local cor-relations. We support our claim by studying their effecton the two-orbital Hubbard model on the honeycomb lat-tice for the moir´e superstructure with a U(1) single-siteslave-spin mean-field technique [21]. We argue that theinclusion of non-local correlations can heal this disagree-ment, indicating that antiferromagnetic-like correlationsplay a key role in the localization of the carriers. II. MODEL
The hamiltonian for the moir´e superlattice comprisesthe kinetic energy H t and the on-site interaction term H U . The tight-binding models for the moir´e superlatticeare not defined in the basis of the atomic p z orbitals ofthe carbon atoms, but on effective orbitals from Wannierprojections for the low energy bands. These effective or-bitals are not defined in a particular graphene layer but inthe TBG. However, the most suitable model is still underdiscussion [8–10, 12–15, 22–26]. Although the AA regionsform a triangular lattice, tight binding models with suchsymmetry spanning only the flat bands do not preservethe Dirac cones [10]. Tight-binding models for differentlattices and number of bands have been proposed.Here we focus on the simplest proposed model whichreproduces the Dirac cones: a two-orbital tight-bindinghamiltonian on the honeycomb lattice [8–11, 22] with or-bitals presumably centered on the AB and BA regionsfeaturing lobes directed toward the AA stacking centers.The energy bands of the two-orbital honeycombmodel [8–11, 22] mimic the flat bands around the chargeneutrality point with bandwidth W ∼
10 meV [6]. Thereare two bands with energy
E >
E < t to first nearest neighbors( t ∼ t . The filling per site in the honeycomb lattice is definedas x = n/ N with n ( N ) the number of electrons (or-bitals) per site. Note that we define the filling from thebottom of the bands, not from the charge neutrality pointas in the original experimental papers [1, 4]. In our nota-tion, the filling at the Dirac points (the charge neutralitypoint of the TBG) corresponds to half-filling x = 1 / x = 1 / n = 1) and x = 3 / n = 3), the density correspondingto the ∓ H U in-cludes the intra- and inter-orbital interactions U and U (cid:48) ,the Hund’s coupling J H , and pair-hopping J (cid:48) terms [8,27]: H U = U (cid:88) jγ n jγ ↑ n jγ ↓ + ( U (cid:48) − J H (cid:88) j { γ>β } σ ˜ σ n jγσ n jβ ˜ σ − J H (cid:88) j { γ>β } (cid:126)S jγ (cid:126)S jβ + J (cid:48) (cid:88) j { γ (cid:54) = β } c † jγ ↑ c † jγ ↓ c jβ ↓ c jβ ↑ (1)with γ and β labelling the orbitals, j the sites and σ, ˜ σ the spin. Here J (cid:48) = J H and U (cid:48) = U − J H [8, 27]. U has been estimated to be ∼
20 meV [1]. The value of theHund’s coupling is not known. Yuan and Fu [8] assume J H = 0 and U = U (cid:48) . On the other hand, Dodaro et al [14]argue that once the effect of the phonons of the TBGwith frequencies ω ∼
200 meV is included the effective J H becomes negative, as for alkali-doped fullerides [28].We do not make any a priori assumption on the value orthe sign of J H . We assume the equality U (cid:48) = U − J H remains valid for J H < [20]or U(1) [21] versions, have proven to be very useful tocalculate the critical value of the interaction U cx . Thepredictions from Dynamical Mean Field Theory (DMFT)and those from slave-spin are compared in Ref. [33] forthe quasiparticle weight and the critical interaction forseveral electronic fillings and values of the Hund’s cou-pling. Very good agreement between both techniques isfound [33, 35]. The slave-spin technique does not giveinformation on the spectral weight redistribution whichtakes place with increasing interactions: Spectral weight FIG. 1: Interaction U cx at which the insulating Mott stateemerges for quarter- ( x = 1 /
4) and half-filling ( x = 1 /
2) as afunction of Hund’s coupling J H in the two-orbital honeycomblattice. U c / < U c / is satisfied only in a very small rangeof J H centered around J H ≈ U c / = 14 . t and U c / = 19 . t . The dotted line corresponds to U / , the criti-cal interaction for the Mott transition in a single-orbital modelin the honeycomb lattice. The inset shows the non-interactingdensity of states as a function of electronic filling. The lowerhorizontal axis shows the number of electrons n per latticesite in the two orbitals and the upper axis shows the elec-tronic filling with respect to its maximum value. Half-filling x = 1 / ± is transferred from the quasiparticle peak to the inco-herent/many body Hubbard bands, producing a threepeak spectral function. We obtain this information fromprevious calculations performed with DMFT [17] in theliterature and cited accordingly in the text. III. COMPARISON WITH EXPERIMENTALOBSERVATIONSA. Doping dependent critical interaction
In a two-orbital model the insulating states can befound at fillings x = 1 / , / /
4. The insulatingstates with ± x = 1 / x = 3 /
4, with 1 and 3 electrons persite (2 and 6 per moir´e unit cell), respectively, with re-spect to the bottom of the band. Due to the particle-holesymmetry of the model, below we restrict the discussionto x ≤ / U c / ∆ U / , leading to U c / < U c / , in disagreementwith the experimental observations. At zero Hund’s cou-pling ∆ U / = ∆ U / = U . On spite of having the samecharging energy cost, as seen in Fig. 1 and previouslydiscussed [8, 16, 19, 36] for small values of J H , orbitalfluctuations stabilise the metallic state up to larger val-ues of U at half-filling than at quarter-filling, such that U c / > U c / , consistent with experiments [8]. Fromnow on, we focus on the small range of Hund’s cou-pling − . < J H /U < .
01 in which the condition U c / < U c / is satisfied. We approximate such smallvalues by J H = 0. We note that due to the larger degen-eracy of the multi-orbital insulating states, these orbitalfluctuations make both U c / and U c / at small J H largerthan U / , the critical interaction for the Mott transi-tion in the single-orbital model in the honeycomb lattice. B. Magnetic field dependence
Experimentally, the insulating state at x = 1 / H ∼ −
6T ( µ B H ∼ . − .
36 meV) [1]. This behaviour does not dependon whether the field is applied perpendicular or parallelto the TBG sample, suggesting that the suppression ofthe insulating state is due to the Zeeman effect. This isconsistent with the fact that for localized electrons theorbital effect is not expected to play a role.Agreement with experiments requires that the criticalinteraction for the Mott transition U c / increases in amagnetic field. However, contrary to the experimentalobservations, we find that in the local approximation amagnetic field decreases U c / promoting the insulatingtendencies, see Fig. 2. The effect of the magnetic field H is introduced via the Zeeman term H (cid:80) jγ ( n jγ ↑ − n jγ ↓ ).At x = 1 /
4, with J H = 0, the atomic gap ∆ U / = U doesnot depend on H . When a magnetic field is applied, theZeeman term favors spin polarization. When the bandsare completely spin polarized (for instance, the spin upband is emptied) the behaviour of the spin down bandbecomes equivalent to that of a single orbital model athalf-filling. As discussed above, the critical interactionfor a single orbital system U / is smaller than U c / at H = 0. The magnetic field lifts the spin degeneracysuppressing the orbital fluctuations. As a consequence, U c / decreases with the magnetic field towards U / .This is shown in Fig. 2. More details are given in theAppendix A. C. Temperature dependence and gap size
The insulator to metal transition observed experimen-tally with increasing temperature T is also at odds withthe behaviour expected from a charge gap emerging fromlocal correlations. In single-orbital Hubbard models, the T dependence of the Mott transition in the paramag-netic phase has been extensively studied using single-siteDMFT [17, 37]. The results are summarized in Fig. 3 (seethe blue line on the right) where U c, local refers to U / .If at low T the system is insulating (for U > U c , local ) theresistivity decreases with increasing T , but the metallicbehaviour is not restored [37]. Below U c, local the systemis metallic and at a critical T there is a first order transi-tion, blue solid line, to an insulating state. This happensbecause in the local approximation the entropy in theMott state is larger than in the metallic one [17, 38].This phenomenology has been experimentally observedin oxides [39] and remains unchanged in the two-orbitalcase [16]. Hence, if only local correlations are included,the insulating behaviour is not suppressed with T , con-trary to the experimental observations.Previous works have pointed out that the observedsmall gap ∼ . ∼ U − W ∼ ( U, W ), with W the bandwidth, as obtained by the single-site DMFTtreatment of the Hubbard model [17].The lack of agreement between the experimental ob-servations (dependence on magnetic field and T , and thegap size) and theoretical predictions including only localcorrelations is restricted neither to the two-orbital char-acter of the Hubbard model nor to particular lattices [41]or tight-binding models. These disagreements indicatethat the experimentally observed insulating state cannotbe described as a Mott state considering only local cor-relations. Below we argue that the inclusion of inter-site correlations in the Mott transitions could be the clue toreconcile theory and experiment. IV. EFFECT OF INTER-SITE CORRELATIONS
Even within the Hubbard model (with only on-siteinteractions) inter-site correlations are generated. Thebest known example is the single orbital case at half-filling for which antiferromagnetic (Heisenberg-like) cor-relations between the local electrons in neighboring sitesemerge. In multi-orbital systems the inter-site correla-tions involve both magnetic and orbital degrees of free-dom, and depend on J H [42, 43] and the filling. Fora two-orbital system with x = 1 / J H = 0 correlations are AFM and antiferroor-bital [46].Previous calculations in several lattices, mostly re-stricted to single-orbital models, have found that includ-ing the effect of short-range inter-site correlations in theMott transition decreases the critical interactions and canconsiderably change the expected dependences [40, 47].Their effect in non-ordered states have been studied withcluster approaches [47] in the context of organic super-conductors and cuprates where they play an importantrole. Unlike local correlations, these effects do depend onthe lattice and become more important when the effec-tive dimension is reduced, as in the honeycomb lattice,with small coordination number z = 3.Cluster treatments of multi-orbital models are compu-tationally challenging and very scarce [42, 43, 46, 48, 49].To our knowledge there are no studies which include thesenon-local correlations in the two-orbital honeycomb lat-tice. However, the expected qualitative behaviour in thenon-ordered Mott state of the present model can be in-ferred from previous results in the two-orbital square lat-tice and in single-orbital models with different lattice ge-ometries.As shown in Fig. 3, non-local correlations shiftthe Mott transition to a smaller critical interaction U c, non − loc [40, 46, 47, 50–52] even in the absence of mag-netic order. The correlated insulator decreases its energywith respect to the correlated metal. While for large T and U the local approximation gives a good descriptionof the electronic properties, at lower temperatures andclose to U c, non − loc (green line on the left in Fig. 3), thebehaviour is controlled by the inter-site correlations. Inthe single-orbital case at half-filling and with AFM cor-relations, the entropy of the insulator decreases due toshort-range singlet formation [40]. As a consequence, thecritical interaction U c, non − loc reverses its slope as a func-tion of T with respect to the local-correlations predic-tion [40, 50, 53] (compare blue and green lines in Fig. 3).Hence, the low T phase is the insulator, as observedexperimentally. Similar behaviour has been found in atwo-orbital model at zero and finite J H [42]. Cluster FIG. 3: Sketch of the temperature-interaction phase diagram for the Mott transition in the paramagnetic single-orbital case,previously obtained in DMFT (only on-site (local) correlations are included, blue lines at the right) and CDMFT (inter-site(non-local) correlations are included, green lines at the left). [17, 40]. Above a given temperature the phase transition turnsinto a crossover (dashed lines). With only on-site correlations, the insulating behaviour is enhanced when the temperature isincreased. For U below the blue line, the metallic state turns into an insulator with increasing temperature. Above the blueline, the system is always an insulator and its resistivity decreases with increasing temperature. When short-range inter-sitecorrelations are included, the critical interaction decreases (green line) and the low T slope reverses due to the short-rangesinglet formation. Non-monotonic behaviour is observed and at sufficiently high-temperature both local and non-local crossoverlines converge into a single one. In the region of parameters between the local and non-local lines, the system is metallic if onlylocal correlations are considered, and insulating with inter-site correlations. In the later case, the system is insulating belowthe green line and metallic above it. The red arrows indicate the effect of a magnetic field on the critical interactions. DMFT (CDMFT) calculations in the paramagnetic two-orbital square lattice at x = 1 / J H = 0 [46] havealso found that the AFM correlations are suppressed bytemperature. With increasing T the crossover betweenthe metal and the insulator is non-monotonic, see Fig. 3,and it matches the crossover line found in the single-siteapproximation at high temperatures (both dashed linesmerge). The qualitative behaviour when non-local cor-relations are included is also found for other lattices. Inparticular, it is expected to be found in the honeycomblattice, bipartite as the square one. This has been alreadystudied in the single-orbital case at half-filling [40, 51]. Alarger degree of magnetic frustration decreases the mag-nitude of the effect, i.e. U c, non − loc becomes closer to thelocal predictions and the range of temperature in whichnon-local correlations control the experimental behaviourgets narrower [54]. We emphasize that the temperaturedependence compatible with experimental observationsis found only very close to U c, non − loc , at interactions forwhich the system is metallic in the absence of inter-sitecorrelations (namely, below the green line in Fig. 3). Thisresult suggests that inter-site correlations play a key rolein the localization and insulating states observed experi-mentally in TBG.The gap size controversy can also be accounted forby the inclusion of non-local correlations. While for lo-cal correlations the quasiparticle peak disappears at theMott transition at U c, local and the gap is the one be- tween the Hubbard bands, for U c, non − loc < U < U c, local (namely, between the green and blue lines in Fig. 3) non-local correlations open a small gap in the quasiparticlepeak for smaller interactions [40, 55]. Close to the tran-sition this gap is much smaller than the one between theHubbard bands and its size would be consistent with ex-perimental observations.Close to U c, non − loc , in the region of parameters wherethe temperature dependence and gap size place the ex-perimental system, the insulating state is controlled bythe inter-site magnetic and orbital correlations. If thesecorrelations are AFM, as it happens for zero or negativeHund [44, 46], a sufficiently large magnetic field couldsuppress them and induce a transition to a metallic state,as observed experimentally. A magnetic field does notsuppress ferromagnetic correlations. As a consequence ifthe inter-site correlations were ferromagnetic a magneticfield could not produce an insulator to metal transition,in disagreement with experiments. This indicates thatthe inter-site correlations which assist the Mott transi-tion have antiferromagnetic character.Finally, we note that non-local correlations can alterthe ratio, discussed above, between the critical interac-tions at quarter- and half-fillings. The critical interactionis controlled by the non-local correlations which havea different effect at each filling, as it becomes evidentby their different ordering tendencies. For example, at J H = 0 the ground state of the honeycomb lattice is avalence bond solid [56] at x = 1 / x = 1 / U c, non − loc1 / < U c, non − loc1 / can be achieved forzero or negative Hund’s coupling in the honeycomb lat-tice. V. CONCLUSION AND DISCUSSION
We have explored the phenomenology of the insulatingstates of twisted bilayer graphene with ± J H ≤ U c, non − loc1 / < U c, non − loc1 / , necessary toreproduce the metallic character of the charge neutral-ity point while the system is insulating at quarter-filling,is fulfilled in the local approximation, but could be af-fected by non-local effects. Whether it is satisfied in thepresence of inter-site correlations should be studied infuture work. An interesting aspect of non-local correla-tions is the possibility of finding pseudogap physics. Lo-cal correlations result in a momentum-independent self-energy[40]. Non-local correlations make the self-energymomentum dependent. These momentum dependentcorrelations have been studied in the single-orbital squarelattice when the system is doped away from the insulatingstate [54, 59–64]. The predicted momentum dependencehas been observed in cuprate superconductors and it is akey signature of the pseudogap phenomenology [65].In this manuscript we have described the TBG witha two-orbital Hubbard model on the honeycomb lattice.The predicted behaviour for the metal insulator transi-tion with temperature and magnetic field, as well as thesmall gap close to the transition, are not restricted to thismodel. We expect that our qualitative conclusions wouldremain valid in other models proposed for the TBG whichfeature Mott transitions at the experimental fillings. Onthe other hand, the range of interaction parameters, tem-perature and magnetic field in which the inter-site cor-relations control the experimental behaviour, the ratio U c, non − loc1 / /U c, non − loc1 / , or the possible pseudogap physics will depend on the model. The description of the insu-lating states with ± ± ± Appendix A: Appendix: Critical Interactions
The critical interaction for the Mott transition dependson the charging energy cost ∆ Ux , see main text. For thepresent model, at half-filling, ∆ U / = U + J H if J H ≥ U / = U + 5 | J H | if J H < U / = U − J H if J H ≥ U / = U if J H <
0. The vanishing density of states at x = 1 / U c / /U c / ∼ .
28 at J H = 0 to slightlylarger values than in other lattices [16, 19, 29]. For in-stance, in the square lattice, with a van Hove singularityat x = 1 / U c / /U c / ∼ .
13. Away from J H = 0, U c / decreases while U c / has a positive slope, see Fig. 1.These dependences reflect the behaviour of ∆ Ux [29].We now focus on J H = 0 and x = 1 /
4. In the metallicstate the strength of the correlations can be quantifiedby the value of the quasiparticle weight Z . Z = 1 inthe non-interacting limit and it decreases with increasinginteractions. At the Mott transition Z vanishes. As seenin Fig. 4(b), in the absence of a magnetic field, Z vanishesat interactions U c / (H=0) larger than U / .In the presence of a magnetic field the bands becomespin polarized and the quasiparticle weight becomes spindependent Z σ . The Mott transition happens when thequasiparticle weight of the majority spin band Z ↓ goesto zero. If the orbitals are completely spin polarized thesystem becomes equivalent to a single orbital model at x = 1 / U = U for J H = 0. Ex-cept for very small H the spin up band is emptied atinteractions U pol < U / , as it can be seen in Fig. 4(a). FIG. 4: (a) Filling per orbital γ for the majority (down) spin n γ ↓ as a function of U for x = 1 / J H = 0, and differentvalues of the magnetic field H . n γ ↑ = 0 . − n γ ↓ . 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