From quantum anomalous Hall phases to topological metals in interacting decorated honeycomb lattices
FFrom quantum anomalous Hall phases to topological metals in interacting decoratedhoneycomb lattices
Manuel Fern´andez L´opez and Jaime Merino
Departamento de F´ısica Te´orica de la Materia Condensada,Condensed Matter Physics Center (IFIMAC) and Instituto Nicol´as Cabrera,Universidad Aut´onoma de Madrid, Madrid 28049, Spain (Dated: May 1, 2019)An analysis of the stability of topological states induced by Coulomb repulsion on decoratedhoneycomb lattices is presented. Based on a mean-field treatment of a spinless extended Hubbardmodel on the decorated honeycomb lattice we show how the quantum anomalous Hall (QAH) phaseis a robust topological phase which emerges at various electron fillings and involves either quadraticband crossing points (QBCP) or Dirac points of the bands. The topological QAH phase is also foundto be most stable against thermal fluctuations up to moderate temperatures when the Coulombrepulsion is maximally frustrated. We show how a topological metal can be induced from the QAHby electron doping the system in a broad electron doping range. Electrons on the Fermi surfaceof such metallic states are characterized by having a non-zero Berry phase which gives rise to anon-zero intrinsic quantum Hall conductivity.
I. INTRODUCTION
Topological insulators are being intensively studiedsince their discovery . In spite of having a bulk gap,they are conducting in the surface with degenerate edgestates crossing the Fermi energy which are protected bytime-reversal symmetry (TRS). The associated topolog-ical bulk invariant, Z , characterizes the Quantum SpinHall (QSH) phase predicted in graphene in the presenceof strong enough spin-orbit coupling (SOC). In systemswith broken TRS, the Quantum Anomalous Hall (QAH)phase which is characterized by a non-trivial topologi-cal invariant, the Chern number , can arise. Very re-cently, both kinds of topological phases have been the-oretically predicted in the decorated honeycomb lattice(DHL). In a tight-binding model of the DHL in the pres-ence of SOC, the QSH phase, characterized by a non-vanishing Z invariant, emerges. When the Coulomb in-teraction is added to the tight-binding model, TRS isspontaneously broken giving way to a QAH phase char-acterized by a non-zero Chern number. This phase canbe associated with the spontaneous generation of finite’magnetic’ fluxes piercing the elementary hexagonal plac-quettes of the lattice but with zero net flux through theunit cell. This phase is analogous to the quantum Hallphase without an applied magnetic field generated byadding complex next-nearest-neighbors (n.n.n) hoppingamplitudes to the tight-binding model on the honeycomblattice. The DHL is interesting not only from the theoret-ical point of view but also because it is realized inactual materials such as the trinuclear organometal-lic compounds e. g. Mo S (dmit) , in Iron (III)acetates or in cold fermionic atoms loaded in a deco-rated honeycomb optical lattice . The hopping param-eters, entering the tight-binding hamiltonian (1), H tb isshown in Fig. 1(a). The DHL can be seen as interpolat-ing between the honeycomb and the Kagom´e lattice. The band structure is richer than on the honeycomb lat- tice potentially leading to novel topological states of mat-ter. Apart from the Dirac points protected by TRS andinversion symmetry (IS), the band structure also displaysquadratic band crossing points (QBCP) which are topo-logically protected by C or C symmetries . The twoQBCP’s involve a flat band which in turn leads to diver-gences in the density of states (DOS) are shown in Fig.1(c) which are relevant for inducing instabilities of thesystem, particularly around f = 1 / f = 5 / K ( K (cid:48) ) so that the systembecomes an insulator. Such insulator is topologicallynon-trivial consisting on some bands with non-zero Chernnumbers ( ν n ) as displayed in Fig. 1(d), implying topo-logically non-trivial states arising at certain electron fill-ings. These states are due to the spontaneous formationof QAH phases induced by the Coulomb repulsion. Thetwo lowest energy bands with Chern numbers: ν , = ± In contrast, the bands n = 4 , ν , = 0, due to the can-cellation of the Berry phase (of ± π ) involving the QBCParound the Γ-point and the Berry phases of ± π associ-ated with the Dirac cones at K and K (cid:48) .In the present work, we study the doping and temper-ature effects on the QAH phase generated by Coulombrepulsion on a DHL. Previous work on the DHL has con-centrated on the non-interacting QSH phase induced bySOC and the QAH phase induced by Coulomb repul-sion but only at half-filling, f = 1 /
2. However, littleis known about the existence of the QAH at other fill-ings and/or as temperature is raised. We cover this gapby obtaining the full phase diagram of an spinless ex-tended Hubbard model with nearest-neighbors (n.n), V ,and n.n.n, V Coulomb repulsion on the DHL at all rel-evant integer fillings and at certain non-integer fillings. a r X i v : . [ c ond - m a t . s t r- e l ] A p r (a) (b)(c) (d) FIG. 1: Band structure of the decorated honeycomb lattice.(a) Unit cell and hopping parameters ( t , t (cid:48) ) of the tight-binding model on a DHL, (b) the first Brillouin zone of theDHL, (c) the band structure of the tight-binding model ona decorated honeycomb lattice for t = t (cid:48) is shown along the k -path M → Γ → K → M . The corresponding density ofstates is plotted at the side. The band structure displaysDirac points at K (and K (cid:48) ) as well as quadratic band cross-ing points (QBCP) at the Γ-point (d) The Hartree-Fock en-ergy bands of the spinless model (1) in the presence of theCoulomb interaction ( V ∼ , V /t ∼ .
35 at f = 1 / K ( K (cid:48) )points making the system insulating at integer fillings. TheChern number of each band is shown on the side of the plot.When doping with electrons off half-filling between f = 1 / f = 2 / K -point. We find topological metallic phases arising at filling frac-tions varying, for instance, between f = 1 / f = 2 / V , V ) parameters for which the QAH isthe ground state. Non-zero intrinsic Hall conductivitiesas well as magnetoresistance oscillation experiments as-sociated with the Berry phases on the Fermi surface canprovide experimental evidence for the existence of suchtopological metals.The paper is organized as follows. In Sec. II we de-scribe the model and the Hartree-Fock method used toanalyze the model. In Sec. III we obtain the ground state phase diagram of the model at the filling fractions, f , at which either QBCP or Dirac points are relevant i. e. for f = 1 / f = 1 / f = 2 / f = 5 / T = 0 providing a T − V phasediagram for specific Coulomb parameters for which theQAH is the ground state. By electron doping the QAHwe show how topological metallic phases arise. Finally,in Sec. IV we discuss the implications of our findings onexperimental observations of the Hall conductivity andmagnetoresistance oscillations. We close up the paperwith the conclusions in Sec. V. II. METHODS
In order to analyze possible topological states emerg-ing from the Coulomb repulsion, we consider a spinlessextended Hubbard model on a DHL: H = H tb + H Coul H tb = − t (cid:88) (cid:104) ij (cid:105)(cid:52) c † i c j − t (cid:48) (cid:88) (cid:104) ij (cid:105)(cid:52)→(cid:52) c † i c j H Coul = V (cid:88) (cid:104) ij (cid:105) n i n j + V (cid:88) (cid:104)(cid:104) ij (cid:105)(cid:105) n i n j (1)where the fermion occupation operator is defined as n i ≡ c † i c i . We consider the Coulomb repulsion betweenelectrons in nearest and next-nearest neighbour sites pa-rameterized by V and V , respectively. Since this hamil-tonian is cuartic in the fermion operators, it cannot besolved exactly so we apply a Hartree-Fock mean-field de-coupling of these terms: n i n j ∼ ( n i n j ) Hartree − ( n i n j ) Fock (2)where ( n i n j ) Hartree = n i (cid:104) n j (cid:105) + (cid:104) n i (cid:105) n j − (cid:104) n i (cid:105)(cid:104) n j (cid:105) and( n i n j ) F ock = c † i c j (cid:104) c † j c i (cid:105) + (cid:104) c † i c j (cid:105) c † j c i − (cid:104) c † i c j (cid:105)(cid:104) c † j c i (cid:105) . Wework in the canonical ensemble with a fixed number ofelectrons N e . At a given temperature β = k B T , the freeenergy F is given by F = F T + F H + F F , where: F T = − k B T (cid:88) k ,n log [1 + e − β ( E k ,n − µ ) ] + µN e (3) F H = − V (cid:88) (cid:104) ij (cid:105) (cid:104) n i (cid:105)(cid:104) n j (cid:105) − V (cid:88) (cid:104)(cid:104) ij (cid:105)(cid:105) (cid:104) n i (cid:105)(cid:104) n j (cid:105) (4) F F = V (cid:88) (cid:104) ij (cid:105) (cid:104) c † i c j (cid:105)(cid:104) c † j c i (cid:105) + V (cid:88) (cid:104)(cid:104) ij (cid:105)(cid:105) (cid:104) c † i c j (cid:105)(cid:104) c † j c i (cid:105) (5)with µ the chemical potential and E k ,n the Hartree-Fockband dispersions. In order to find the mean field ampli-tudes that minimize the free energy we solve the followingsystem of coupled equations: ∂ F ∂ (cid:104) n i (cid:105) = 0 (6) ∂ F ∂ (cid:104) c † i c j (cid:105) = 0 (7) ∂ F ∂ (cid:104) c † j c i (cid:105) = 0 (8)The number of variables of the whole system is 27 (6 (cid:104) n i (cid:105) and 21 (cid:104) c † i c j (cid:105) ). Since finding the global minimum of F isnot a straightforward task, we first fix V ∼ V (cid:28) t andsolve the equations at the filling fraction, f = 1 /
2. Wecan see that the converged solutions display the followingpattern in the Fock terms: (cid:104) c † i c j (cid:105) ≡ χ = ξ + iη (cid:52) st (cid:104) c † i c j (cid:105) ≡ χ = ξ + iη (cid:52) → (cid:52) st (cid:104) c † i c j (cid:105) ≡ χ = ξ + iη nd (9)where (cid:52) and (cid:52) → (cid:52) mean the intratriangle and in-tertriangle neighbours, respectively. In actual calcula-tions, we reduce our system of equations by fixing this ansatz . We observe that the Fock amplitudes that mini-mize the energy are complex χ m ∈ C and the mean den-sities at each site (cid:104) n i (cid:105) are all the same at this filling. Thismeans that time-reversal symmetry is spontaneously bro-ken while the rotational C symmetry is preserved. Thisphase is a quantum anomalous Hall (QAH) phase andit is characterized by the presence of finite fluxes ( η m )through elementary placquettes of the lattice (see Fig.3(b)). However, the total flux through the unit cell iszero due to the periodic boundary conditions. The Fockcontribution to the total Hartree-Fock free energy (5) as-suming this ansatz takes the form: F F = 6 V ( ξ + η ) + 3 V ( ξ + η ) + 12 V ( ξ + η )(10)Looking at the free energy expressions (4) and (10), theequations for the densities (6) and for the Fock ampli-tudes (7) seem to be decoupled. However, due to the de-pendence of the band dispersions E k ,n ( (cid:104) n i (cid:105) , ξ m , η m ) onboth the local densities and the Fock amplitudes enter-ing the thermal part (3), they form a set of coupled self-consistent equations: V (cid:88) j ∈(cid:104) ij (cid:105) n j + V (cid:88) j ∈(cid:104)(cid:104) ij (cid:105)(cid:105) n j = (cid:88) k ,n ∂E k ,n /∂n i e β ( E k ,n − µ ) ξ m = − a m V , (cid:88) k ,n ∂E k , n /∂ξ m e β ( E k ,n − µ ) η m = − a m V , (cid:88) k ,n ∂E k , n /∂η m e β ( E k ,n − µ ) (11) where i is the sites index (1 ≤ i ≤
6) and m is theFock ansatz index (1 ≤ m ≤ j ∈(cid:104) ij (cid:105) and j ∈ (cid:104)(cid:104) ij (cid:105)(cid:105) refer to the n.n and n.n.n of eachsite i , respectively. a m are coefficients which come fromthe derivation of (10), and V , is either V or V for m = 1 , m = 3 respectively (9). The equationsare written in a compact way: six equations in the firstline corresponding to each site i while in each restingones (second and third line), three equations are showncoming from each ansatz index, m . This system of 12equations can be solved iteratively.Although the system has been simplified through the ansatz assumed, in some situations the convergence toone or other minima may depend on the initial guessseeded into the equations. Hence, we proceed as follows.For a given set of ( V , V ), we search for the global mini-mum by comparing the free energy of the complete set ofequations (11) with the free energies obtained with thesame set of equations satisfying additional constraints.In this way we can reduce the unwanted dependence ofthe solution on the initial guess plugged into the sys-tem of equations. The first constraint imposed on theequations consists on assuming a pure Hartree decouplingwhich fixes χ m = 0, so that the system of equations isrestricted to the (cid:104) n i (cid:105) only giving the different possiblecharge ordering patterns of the model. The second con-straint consists on imposing Fock amplitudes which arepurely imaginary, ξ m = 0, giving the contribution to thechiral currents. The third and final constraint restrictsthe system to solutions of the type: η m = 0, so that the χ m , being real, describe shifts of the hopping parame-ters δt m : t ( (cid:52) ) → − t + V δt , t ( (cid:52) → (cid:52) ) → − t (cid:48) + V δt , t (2 nd ) → V δt . As stated above, the free energies F are evaluated separately assuming each of the threeconstraints described above and compared to full set ofequations without any constraint imposed. The solutionwith the lowest free energy provides the ground state fora fixed set of ( V , V ). This allows to construct the phasediagrams of Fig. 2.The topological properties of the insulating solutionsare characterized by calculating the total Chern num-ber ( ν ) of the system. The QAH phase is characterizedby having a non-zero Chern number ( ν (cid:54) = 0) in analogywith the standard Quantum Hall effect as shown in Fig.3(b). The Chern numbers are evaluated numerically byevaluating the Berry flux through the elementary plac-quettes in which the first Brillouin zone is discretizedwhich neutralizes the arbitrary phase coming from thegauge invariance. The topological properties of metallicstates are analyzed in a similar way by computing theBerry flux through the F S defined by the partially filledbands. This gives the non-quantized contribution to theintrinsic Hall conductivity σ xy . The details regardingthese numerical procedures are given in Appendix A.
III. RESULTS
In this section, we first explore the ground state of themodel for different ( V , V ) following the procedure de-scribed above at each filling fraction, f . Then we checkthe stability of the QAH phase against thermal fluctu-ations. This is done for the fixed V /V ratio at whichthe QAH is the ground state of the model. We solve theequations at finite temperature, obtaining a T − V phasediagram. This allows to extract an estimate of the tem-perature below which the QAH phase is stable. Finally,we electron dope the system varying the filling between f = 1 / f = 2 / V , V , at which the QAHstate is the ground state of the system. In this way westudy the evolution of the Fermi surface and the possibletopological metallic states emerging in the model. A. Ground state phase diagram
We first discuss the ground state phase diagram ofmodel (1). We obtain the phase diagrams at four fillingfractions, f , at which the conduction and valence bandshave either Dirac or quadratic band touching points. Wesearch for the ground state of the model for given V , V following the method explained in the previous section FIG. 2: The V − V phase diagrams at various filling fractions, f . (a) At f = 1 / f = 1 / f = 2 / f = 5 / (see Fig. 2).The resulting phase diagrams are shown in Fig. 2.We find uniform charge density (UCD), Charge DensityWave (CDW) and Nematic Insulator (NI) phases. Whilethe uniform charge density (UCD) state preserves thesymmetries of the lattice, the CDW phase breaks the C rotational invariance of the lattice reducing it to a mirrorsymmetry only. The reflection plane goes through eitherintratriangle (in CDW I and CDW I* phases) or inter-triangle (in CDW II) bonds. These states are three-folddegenerate. Although the charge distribution in the unitcell changes by applying 60 rotations, the total energyis invariant. There are different types of charge orderingpatterns in these phases as shown in Fig. 3. The NIphase consists of an almost empty triangle and the restof the charge uniformly distributed in the other triangleof the unit cell. This state is two-fold degenerate due (a) (b) (c)(d) (e) (f) FIG. 3: The different ground states shown in the V − V phasediagrams of Fig. 2. (a) Uniform charge density (UCD). Thecharge density is uniformly distributed in the unit cell andthe bonds ξ m (cid:54) = 0 follow the ansatz (9) preserving C symme-try. (b) Quantum anomalous Hall (QAH) phase. This phasepreserves C symmetry while TRS is broken by spontaneouschiral currents η m (cid:54) = 0 being two-fold degenerate dependingon the direction of the current (the represented correspondsto a Chern number, ν = +1). (c) Nematic insulator (NI). Inthis state the charge inside the unit cell is located in one ofthe triangles of the unit cell. The rotational symmetry is re-duced from C to C leading to a two-fold degenerate groundstate. (d) Charge density wave I (CDW I). The charge is dis-tributed following the colour patterns displayed. Inside eachunit cell nearest-neigbor sites are paired up with the samecharge. (e) CDW I*. In this phase the inter-triangle nearest-neighbours sites have different densities. (f) Charge densitywave II (CDW II). The densities are associated in pairs butnot between nearest-neighbors (except for the n. n. intertri-angle sites). In the CDW phases the hamiltonian is invariantunder reflection transformations: with respect to the x -axisfor CDW I and CDW I*, and with respect to the y -axis forCDW II. All these states are three-fold degenerate. to the reduction of rotational symmetry from C downto C . The transition line separating this phase fromthe CDW or QAH phases is second order as shown inFig. 2. When ∼
90% of the total charge in a unit cell islocalized in one of the triangles we assume that the tran-sition has occurred. With four electrons per unit cell, f = 2 /
3, three electrons are located in one triangle whilethe electron left distributes uniformly among the sites ofthe other discharged triangle. We denote this phase byNI*.Comparing the phase diagrams at different fillings, weconclude that the QAH phase is robust occurring at allfilling fractions except for f = 1 /
6. We can rational-ize this from the fact that, at this filling, there is onlyone electron per unit cell so that the effective Coulombrepulsion is not strong enough to destabilize the semimet-alic (SM) phase and turn it into the QAH phase. Thishas also been found in the Kagom´e lattice, in which theemergence of the QAH phase requires a third neighbourinteraction. However, at f = 5 / f = 5 / f = 1 / C ro-tational symmetry but the QAH breaks TRS η m (cid:54) = 0, asshown in Fig. 3. At filling fractions involving QBCP’s i.e. at f = 1 / / f = 1 /
2, along the line V /t ∼ V /t , the QAH phase is the most stable. This meansthat the energy difference with the competing UCD statereaches its maximum at this filling (see Fig. 7(a) in Ap-pendix B). Hence, we choose this range of parameterswith f = 1 / B. Finite temperatures
As stated above we analyze the effect of temperatureon the QAH phase at f = 1 / V ≡ V /t ∼ V /t .Our mean-field T − V phase diagram is shown in Fig. 4.In the phase diagram temperatures are given in Kelvinusing the hopping parameter t = 0 .
05 eV correspondingto Mo S (dmit) crystals. In the limit T → V (cid:46)
2, abovethis value the transition to the CDW I occurs. Subse-quently around V ∼ .
1, a second order phase transi-tion to the NI phase occurs. Observe in the phase dia-gram how thermal fluctuations induce a transition from
FIG. 4: T − V phase diagram at filling fraction f = 1 / V = V /t ∼ V /t until V ∼ C symmetry except for 1 . 84 K. Thehopping of t = 0 . 05 eV is taken for obtaining temperatures inKelvin. a QAH phase to a UCD for 0 < V < . . < V < 2. In the former case the C rotationalinvariance of the lattice is preserved across the transi-tion whereas the latter transition involves a spontaneousbreaking of the rotational symmetry of the lattice. We donot find any further charge ordering transitions beyondthe temperature range shown in the figure implying therobustness of the NI and CDW I against thermal fluc-tuations. From our phase diagram we conclude that themaximum temperature at which we find an stable QAHphase is: T ∼ 84 K for V ∼ V ∼ . t . In Fig. 7(b) ofappendix B we compare the free energies of the UCD andQAH phases. Based on our mean-field theory analysis weconclude that the QAH phase may be most likely foundin half-filled isolated layers of Mo S (dmit) in a broadrange of temperatures if the V and V parameters aretuned through the optimal V ∼ V ∼ . t values. How-ever, it remains to be seen whether other phases differentto the QAH phase become the ground state. C. Topological metals We now explore the possibility of stabilizing a topo-logical metal by doping the QAH state. By raising theFermi level above zero we partially fill the fourth band inFig. 1(b) without closing the band gaps. This leads to atopological metallic state with broken TRS since the chi-ral currents giving rise to the QAH at f = 1 / f = 1 / f = 2 / f = 2 / f = 5 / f = 1 / f = 2 / V ∼ . V ∼ . 2. For these parameters, FIG. 5: Topological metal arising at non-integer filling frac-tions. The non-interacting Fermi surface (red line) is com-pared to the interacting one for V ∼ . t and V ∼ . t andthree different fillings between f = 1 / f = 2 / 3. TheBerry phase of the partially filled fourth band, γ , is showntogether with the occupancy of the fourth band, N . TheFermi surface evolves from a single particle-like loop enclos-ing the QBCP at the Γ-point and with Berry phase: γ ∼ − π to two hole-like loops around the Dirac points at K and K (cid:48) ,with Berry phases: γ ( K )4 = γ ( K (cid:48) )4 ∼ π giving a total Berryphase of γ ∼ π associated with this band. The change inthe Fermi surface occurs for N (cid:38) . the system is deep in the QAH phase as shown in Fig. 2.We compare the interacting Fermi surface with the non-interacting one corroborating Luttinger’s theorem whichstates that the area enclosed by the FS should not varyas we increase the Coulomb interaction. Luttinger’s the-orem should be satisfied since our calculations are basedon a mean-field treatment of the Coulomb interaction.However, the FS can be deformed by the interaction dueto the non-local nature of the Fock contribution. A sim-ilar deformation of the FS is obtained from the HF ap-proximation to the self-energy, Σ( ω, k ), but on the squarelattice. The topological properties of the metallic state gene-rated from the QAH can be investigated by computingthe Berry phase associated with the partially filled fourthband (A3), γ . As shown in Fig. 5, by weakly dopingthe half-filled system with electrons, the FS consists ona single particle-like loop around the Γ-point. The areaof the FS coincides with the number of doped electronsin the system off half-filling quantified by N . Since thespontaneously broken TRS ground state is two-fold de-generate, we have two possible directions of the chiralcurrents. For the chiral currents shown in Fig. 3(b), theBerry phase is approximately, γ ∼ − π . As the Fermilevel is further increased injecting more electrons in thesystem, the area of the FS increases as expected, withthe same Berry phase as shown in Fig. 5(a). However,above N ∼ . 67 the FS is splitted into two hole-like loopsaround the Dirac points at K and K (cid:48) Fig. 5(c). Preciselyat this point the total Berry phase of the partially filledfourth band, γ , changes sign and becomes γ ∼ π . This is because each loop around K and K (cid:48) contributes thesame positive Berry phase: γ ( K )4 = γ ( K (cid:48) )4 ∼ + π . Thisalso explains why the Chern number of the n = 4 bandis ν = 0 as shown in Fig. 1.We have followed the same procedure for analyzingthe emergence of topological metallic phases in the range f = 2 / → / 6. Up to an occupancy of the fifth bandof N ∼ . 18, the FS consists of two electron-like loopsaround K and K (cid:48) . The FS becomes a single hole-likeclosed loop around the Γ-point at larger doping. Increas-ing the electron doping further, the area of this loop de-creases until it disappears when the fifth band becomescompletely filled; at this point the hole occupancy of thefifth band: N ( h )5 = 1 − N → γ ( K )5 = γ ( K (cid:48) )5 ∼ − π until N ∼ . 18 where the Berry phase changes from γ ∼ π to γ ∼ π . Note that the Berry phases obtained areclose to but not exactly 2 π , and contribute to the non-quantized part of the intrinsic Hall conductivity , σ xy ,as discussed below. IV. DISCUSSION We now discuss the implications of our results on ex-perimental observations. The intrinsic Hall conductivity σ xy has two contributions: one coming from the quan-tized part of the N C occupied bands with Chern numbers( ν n ) and the other coming from the nonquantized partassociated with the Berry phase of the FS of the partiallyfilled band, γ N C +1 , reading: σ xy = − N c (cid:88) n =1 ν n − e h γ N c +1 π (12)where N c is the number of totally occupied bands and N c + 1 is the partially filled band. In Fig. 6 we showthe dependence of the Hall conductance on the electrondoping which we quantify through the occupancies of thefourth and fifth band, N , . At half-filling, N c = 3, thesystem consists of three filled bands. Since the two low-est bands have opposite Chern numbers and the thirdband has a Chern number ν = +1 (see Fig. 1), thesystem has a net Hall conductivity of σ xy = − e h whichsurvives as the Fermi energy is moved across the gapas shown in Fig. 6 (around the origin corresponding to f = 1 / γ = − π , contributes to σ xy giving a total con-ductivity, σ xy = 0. However, if we keep increasing theFermi energy there is a sudden change of the Fermi sur-face occurring around N = 0 . 67, at which the Berryphase γ ∼ π . This should be added to the − e h contri-bution of the three lowest filled bands giving a total Hallconductance: σ xy ∼ − e h . Hence, around N ∼ . FIG. 6: Intrinsic Hall conductivity, σ xy as a function of thefilling factor at V ∼ . t and V ∼ . t . The gapped zone ateach filling fraction, f , the Hall conductance is constant andequal to σ xy = − e /h . In the gapless region, with the bands n = 4 (on the left) and n = 5 (on the right) partially filled, ajump in the conductance is observed. These jumps occur justwhen the FS changes from one to two loops and from two toone loop at N ∼ . 67 and N ∼ . 18 respectively. The jumpis about (cid:46) e /h . The finite N , range in which the jumpoccurs is just where the FS is not a closed path. we expect a jump in the Hall conductivity from 0 to ∼ − e /h ) as shown in Fig. 6. A similarjump of the Hall conductivity is found between f = 2 / f = 5 / f = 1 / / 3. This implies that a topological metalemerges. This topological metal is protected by C ro-tational symmetry since the chiral currents induced atthe Hartree-Fock level do not break this symmetry. Itis well-known that the Berry phase can manifest itselfin metals through magnetic oscillatory phenomena. Semi-classical quantization of electron energy levels leadsto the magnetoresistance:∆ R xx ∝ cos[2 π ( B F B + 12 + γ )] , (13)where B F is the frequency of the oscillation associatedwith the area of the electron orbit and γ the Berry phase(in units of 2 π ) picked up by an electron when goingaround it. For instance, using these measurements, aBerry phase of γ = 1 / Electrons in our topolog-ical metal with the FS with closed loops around K andK’ as shown in Fig. 5 would also lead to a Berry phaseshift of γ = 1 / γ → 1. This case would be essentially in-distinguishable from a topologically trivial metal since itcorresponds to an overall shift of 2 π in the magnetoresis-tance oscillations described by (13). V. CONCLUSIONS In the present work we have analyzed, at the mean-field level, the stability of the topologically non-trivialQAH phase induced by offsite Coulomb repulsion on adecorated honeycomb lattice at different fillings and tem-peratures. The QAH phase occurs when band touchingpoints between the non-interacting valence and conduc-tion bands exist. Since the band structure of the deco-rated honeycomb lattice contains both QBCP and Diracband touching points it allows analyzing different topo-logical states arising from them. This is interesting sinceDirac points have Berry phases of ± π associated withthem while the QBCP have a Berry phase of ± π . Inthe presence of the Coulomb repulsion a gap can open uparound these band touching points. This occurs not onlyat f = 1 / 2, as previously reported, but also at f = 2 / f = 5 / f = 1 / 2. Also theQAH is more favorable at f = 5 / f = 1 / f = 2 / V , V ) in which the QAH phase is stable at f = 2 / f = 1 / 2. We believe this isdue to the larger effect of charge frustration due to V at f = 1 / C rota-tional symmetry and spontaneously broken TRS whicharises in our mean field treatment due to non-zero imag-inary Fock amplitudes. The QAH phase competes withthe CDW’s and NI which are topologically trivial insulat-ing phases with the lower reflection and C symmetries,respectively instead of the higher six-fold symmetry. Itis even possible to find other charge ordering patterns. Based on our mean-field treatment we have also analyzedthe stability of the QAH phase with temperature find-ing that the QAH phase at f = 1 / T ∼ 90K for the optimum choice of Coulomb parame-ters: V ∼ V ∼ . t . This temperature can be taken asan overestimation of the actual critical temperature sinceit is based on a mean-field decoupling of the Coulomb in-teraction.We have finally addressed the question of whether atopological metal can be induced by doping the QAHphase. We have explored this by injecting electrons inthe f = 1 / f = 1 / f = 2 / f = 2 / f = 5 / 6. This means that the chiral currents giving riseto the QAH are robust also at such partial filling frac-tions. Our analysis of the Berry phases of the partiallybands indicate that a topological metal with a non-zeroquantized Hall conductivity of (cid:46) − e /h can occur closeto f = 2 / f = 2 / f = 5 / 6. Thenon-zero Berry phases of π arising when the occupancyof the fourth band, N > . 67, could be detected throughmagnetic oscillatory experiments.Future work should include the spin degeneracy in themodel and go beyond the present mean-field treatmenttaking into account electronic correlations in order tocheck whether the QAH phase and the topological metalfound here remain stable. Recent work on the spinfulhalf-filled Hubbard model (no offsite Coulomb repul-sion) on the decorated honeycomb lattice finds a Mottinsulator transition between a semimetallic phase and anantiferromagnetic insulator going through an unconven-tional nematic metallic phase. Both the semimetallic andnematic phases are examples of non-trivial phases emerg-ing from the Coulomb repulsion deserving further char-acterization. Acknowledgments We acknowledge financial support from (MAT2015-66128-R) MINECO/FEDER, Uni´on Europea. Appendix A: Berry phases and Chern numbers inmultiband systems In a discretized Brillouin zone, the total Chern number ν can be calculated from the Berry phases γ nl at eachelementary placquette l . The Berry phase is just theaccumulated phase of the wave function along a certainclosed k -path. γ nl = Im ln N − (cid:89) j =0 (cid:10) u nk j (cid:12)(cid:12) u nk j + (cid:11) (A1)where n is the band index and the loop chosen is rect-angular with N = 4. In the multiband case the wavefunctions overlap of all possible combinations must betaken into account. So that, if we have N c ≡ f valencebands, we construct a N c × N c matrix at each step ofthe path. Then, the Berry phase is just the phase of thedeterminant of the product of these matrices along theloop. The Chern number is nothing else than the sumover the F BZ of all those Berry phases: ν = 12 π (cid:88) F BZ Im ln det (cid:89) j =0 (cid:10) u mk j (cid:12)(cid:12) u nk j + (cid:11) (A2)where 1 ≤ m, n ≤ N c . The shortest steps the closerto an entire Chern number ( ν ∈ Z ). If the band ispartially filled, for instance by doping the material, the system turns into a conductor. If the Fermi surface (FS)is a simple closed loop, the Berry phase of the metal isdetermined at this path. Then in (A2) we restrict thesum to the enclosed surface (FS): γ n = 12 π (cid:88) F S Im ln (cid:89) j =0 (cid:10) u nk j (cid:12)(cid:12) u nk j + (cid:11) (A3)where n corresponds to the partially filled band. ThisBery phase do not have to be a multiple of 2 π contribut-ing to the nonquantized part of the intrinsic Hall conduc-tivity σ xy . Appendix B: Stability analysis of the ground state In the mean-field treatment used in the paper we findthat for a given set of parameters ( V , V ), there are sev-eral solutions having very close free energies. FIG. 7: (a) Dependence of the free energy difference betweenthe two lowest energy states for V /t = V /t ≡ V and f = 1 / η m ≡ η m (cid:54) = 0. In the range 0 (cid:46) V (cid:46) . 9, the freeenergy difference is positive meaning that the QAH phase isthe ground state. It presents a maximum at V ∼ . 4. (b)The UCD and QAH energies as a function of T . The QAHis the ground state up to T ∼ K where the UCD becomesthe lowest energy phase. In Fig. 7(a) we show the free energy difference betweenthe UCD and the QAH at f = 1 / V /t = V /t ≡ V in the T → V ∼ . . t below theUCD. This is the maximum energy difference between theUCD and the QAH as V is increased which is consistentwith the maximum T at which the QAH phase survivesaround V ∼ . V ∼ V ∼ . t . 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