Enhanced compressibility due to repulsive interaction in the Harper model
EEnhanced compressibility due to repulsive interaction in the Harper model
Yaacov E. Kraus,
1, 2
Oded Zilberberg, and Richard Berkovits Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel Institute for Theoretical Physics, ETH Zurich, 8093 Z¨urich, Switzerland
We study the interplay between repulsive interaction and an almost staggered on-site potential inone-dimension. Specifically, we address the Harper model for spinless fermions with nearest-neighborrepulsion, close to half-filling. Using density matrix renormalization group (DMRG), we find that,in contrast to standard behavior, the system becomes more compressible as the repulsive interactionis increased. By generating a low-energy effective model, we unveil the effect of interactions usingmean-field analysis: the density of a narrow band around half-filling is anti-correlated with theon-site potential, whereas the density of lower occupied bands follows the potential and strengthensit. As a result, the states around half-filling are squeezed by the background density, their bandbecomes flatter, and the compressibility increases.
PACS numbers: 71.23.Ft, 73.21.Hb, 73.23.Hk, 37.10.Jk
There has been much interest in the influence ofelectron-electron (e-e) interactions on the compressibil-ity of electronic systems. This interest is motivated bythe intricate many-body physics revealed by the behav-ior of the compressibility, as well as by the technologicalchallenge of building field effect transistors with largercapacitance, essential for lower power consumption andquicker clock rates [1, 2].The compressibility of an electronic system, i.e. thechange in the number of electrons residing in a systemas the chemical potential is varied, can be measured viacapacitive coupling to another metallic system. Alterna-tively, the system can be weakly coupled to a plunger gateand leads. Jumps in the current that passes through theleads as a function of the gate voltage count the numberof electrons in the system as a function of the chemicalpotential. In the context of quantum dots, this is knownas the addition spectrum [3].Compressibility is relevant also for other highly con-trolled many-body systems such as molecular manipula-tion on metal surfaces [4] and ultracold atoms and ions inoptical lattices [5–8]. In optical lattices, transport mea-surements are challenging. Nevertheless, squeezing thetrapping potential acts on the density as a variation ofthe chemical potential, revealing the bulk compressibil-ity [9].Usually, the influence of repulsive interactions on theground state of fermionic systems is well described by amean-field theory, which may be reduced to the classicalcapacitance of the system. Increasing the repulsive inter-action then corresponds to reduced capacitance, i.e. thesystem becomes less compressible. This is true even be-yond the mean-field treatment, as shown for 1D Luttingerliquids with
K < increases with weak e-e in- teraction. Specifically, we study the Harper (or Aubry-Andr´e) model [13, 14] of spinless fermions close to halffilling with nearest-neighbor repulsive interaction. Theon-site potential is spatially modulated with a frequencyof an almost two lattice-sites period, corresponding toa fast modulation with a slow envelope. Using DMRG,we numerically extract the inverse compressibility as afunction of the density, and find it decreasing with theinteraction strength. We analytically show that this ef-fect results from the presence of a flat band at half-filling,which is composed of a superlattice of states that resideat the valleys of the potential envelope. The repulsive in-teraction from occupied lower bands squeezes these valleystates, and accordingly, the central band flattens and itscompressibility increases.The tight-binding Harper model for spinless fermionswith nearest-neighbor repulsive interaction is H = L (cid:88) j =1 (cid:104) tc † j c j +1 + h.c + λ cos(2 πbj + φ ) n j + U n j n j +1 (cid:105) , (1)where c j is the single-particle annihilation operator atsite j , n j = c † j c j is the density, t is a real hopping am-plitude, λ > U > b , and φ is an arbitrary phase factor.The Harper model is a wellspring of physical phe-nomena, and is therefore under continuous study. Forexample, when the modulation frequency b is an irra-tional number, and in the absence of interaction, a metal-insulator transition takes place as a function of the poten-tial strength at λ = 2 t [6, 14–19]. Much effort has goneinto understanding the influence of interaction on thistransition [20–23]. Recently, it was also found that for anirrational b , the Harper model is topologically nontrivial,and may have topological boundary states [24–27]. a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov Here, we are interested in the effect of the interplaybetween the inhomogeneous potential and the interac-tion on compressibility. Therefore, in the following, weassume that we are in the metallic phase, i.e. λ < t .Moreover, we consider the cases of b mod 1 = 1 / (cid:15) with | (cid:15) | (cid:28) /
2, be it rational or irrational. The strikingproperty of such b is that in the vicinity of half-filling,the energy spectrum is composed of an almost-flat cen-tral band separated from the other bands by large gaps,as depicted in Fig. 1(a). Therefore, even weak interactionmay generate interesting phenomena.The inverse compressibility of a system with N par-ticles, ∆ ( N ), is defined as the change in the chemicalpotential due to the insertion of the N th particle. Formany-body systems, it is given by∆ ( N ) = E ( N ) − E ( N −
1) + E ( N − , (2)where E ( N ) is the system’s many-body ground-state en-ergy with N particles. For noninteracting systems at zerotemperature, ∆ ( N ; U = 0) = E N − E N − , where E N isthe N th single-particle eigenenergy.A finite sized Harper model can be thought of as aquasi-disordered 1D quantum (anti-)dot. At low temper-atures, the inverse compressibility of a disordered quan-tum dot is usually described by the CI model, whichhas been shown to fit experimental measurements verywell [3]. According to this model, ∆ ( N ) = ∆ ( N ; U =0) + e /C , where C ≈ L is the total capacitance, and e ≈ U . Thus, an increase in U increases ∆ .We extract ∆ ( N ) of our interacting system usingDMRG [28, 29]. We choose b = √
30 and φ = 0 . π .The former corresponds to (cid:15) ≈ − . L = 200, with N = 91 , , . . . ,
108 elec-trons. For t = 1, the potential amplitude is λ = 0 . greater than the numerical accuracy. Interactionstrengths of U = 0 . , . , . E ( N ) for each N . The numericallyobtained ∆ ( N ), using Eq. (2), is depicted in Fig. 1(b).The accuracy of ∆ drops as U increases, and is about ± · − t for U = 0 .
3. Strikingly, the inverse compress-ibility decreases with increasing U . This implies that theunderlying physics is very different than the one of theCI model.Remarkably, we can reproduce this behavior analyti-cally. First we study the noninteracting case, i.e. U =0. The on-site cosine modulation can be rewritten as λ cos(2 πbj + φ ) = λ cos(2 π(cid:15)j + φ )( − j . Since (cid:15) (cid:28)
1, thepotential is locally oscillating with modulation frequency b = 1 /
2, while being subject to an amplitude envelope, λ ( j ), varying slowly in space with wavelength 1 /(cid:15) , seeFig. 2(a).We postulate that the low-energy physics around E =0, and in particular that of the central flat band, is FIG. 1.
The effect of interaction : (a) The single-particlespectrum of the Harper model [cf. Eq. (1) with U=0] with anopen boundary condition, for L = 200, t = 1, λ = 0 . φ =0 . π , and b = √
30, which corresponds to (cid:15) ≈ − . U . governed by states that minimize both the kinetic andpotential energies. The potential energy is minimizedby states that reside within the valleys of the poten-tial, where the envelope vanishes, i.e. in the vicinity of j ≈ l z , where 2 π(cid:15)l z + φ ≈ ( Z + 1 / π . Within the z th valley of the potential, we can linearly approximatethe envelope, cos(2 π(cid:15)j + φ ) ≈ π | (cid:15) | ( j − l z ) s z , where s z = − sign [sin(2 π(cid:15)l z + φ )] = ±
1. The effective Hamilto-nian for a particle confined to the valley is therefore H valley = L (cid:88) j =1 (cid:104) tc † j c j +1 + h.c. + s z π | (cid:15) | λ ( − j ( j − l z ) c † j c j (cid:105) = (cid:90) π dk π/L ψ † k [2 t cos( k ) σ x + s z π | (cid:15) | λ (ˆ p k − l z ) σ z ] ψ k , (3)where ˆ p k = i∂ k , σ i are Pauli matrices, and ψ k = ( c e k , c o k ) T is the sublattice psuedospinorthat splits the lattice into even and odd sites, ac-cording to c e k = (cid:112) /L (cid:80) L/ j =1 e ik j c j and c o k = (cid:112) /L (cid:80) L/ j =1 e ik (2 j − c j − .Around zero kinetic energy, we linearize cos( k ) ≈− ( k − π/ T = (1+ is z σ x ) / √ H valley = √ tξ (cid:90) π dk π/L ( ˆ T ψ k ) † (cid:18) a † k a k (cid:19) ( ˆ T ψ k ) , (4) FIG. 2.
Modulated potential : (a) The on-site potential (reddots) is a product of a fast alternating part and a slow enve-lope (solid gray line), corresponding to cos(2 π(cid:15)j + φ ). Inset:at its valleys, the potential is linearly approximated. (b) Thedensity of the central band (green), and the background den-sity of the occupied states below it (blue). The filling corre-sponds to numerical results, whereas the dots correspond tothe analytical expressions [cf. Eqs. (5) and (9), respectively].The central band is composed of waves of hybridized Gaus-sians that form a superlattice. Remarkably, the states of thecentral band reside in the potential valleys, whereas the back-ground density follows the potential peaks. where a k = − ( k − π/ ξ/ √ i (ˆ p k − l z ) / √ ξ , and ξ = t/ ( πλ | (cid:15) | ). Since a k satisfies [ a k , a † k ] = 1, it is a lad-der operator. Remarkably, this momentum-space Hamil-tonian is similar to that of the 2D massless Dirac equa-tion in the presence of a perpendicular magnetic fieldin Landau gauge. Using the ladder operators, we findthat the energy spectrum of H valley is ±√ nt/ξ , where n = 0 , , . . . [31]. In particular, there is a zero-energysolution with eigenstate | l z (cid:105) ≈ ( πξ ) − / (cid:80) Lj =1 ( s z ) j S j e − ( j − l z ) / ξ | j (cid:105) , (5)where | j (cid:105) = c † j | vacuum (cid:105) , S j = √ jπ/ − π/
4) = . . . , , , − , − , , , . . . , and we used the fact that ξ (cid:29) ξ around l z . Notably, the wave functions of the excitedstates are also confined with the same Gaussian, similarto the eigenstates of the harmonic oscillator [31].Turning back to the original noninteracting Hamilto-nian, there is a superlattice of valleys, each with its cor-responding zero-energy state. We expect these states tohybridize and form the central band. The | l z (cid:105) statesform a basis for this subspace, since (cid:104) l z | l z ± (cid:105) = 0 and |(cid:104) l z | l z (cid:48) (cid:105)| ≤ e − ( l z − l z (cid:48) ) / ξ (cid:28)
1. We can therefore project the Hamiltonian to this subspace. The projected Hamil-tonian is given by the matrix elements (cid:104) l z | H ( U = 0) | l z (cid:48) (cid:105) .The diagonal elements z (cid:48) = z vanish, since | l z (cid:105) is of zeroenergy. The Gaussian decay implies that for | z (cid:48) − z | ≥ (cid:104) l z | H ( U = 0) | l z ± (cid:105) , namely, hopping betweenneighboring valleys. The resulting effective Hamiltonianfor the central band is [31] H central = − ¯ t (cid:80) L z z =1 ( − z c † l z c l z +1 + h.c. , (6)where L z = (cid:98) | (cid:15) | L (cid:99) is the number of valleys, and ¯ t ≈ e − ξ / (4 ξ (cid:15) ) (cid:0) te − / ξ sinh[(4 ξ | (cid:15) | ) − ] − λe − π (cid:15) ξ (cid:1) .Notably, we obtain ¯ t ≈ . t ≈ . . ξ for ξ in the expression of ¯ t , correctsthe bandwidth.For a periodic boundary condition, the eigenstates of H central are plane waves | k (cid:105) = L − / z (cid:80) L z z =1 S z e ikz | l z (cid:105) with spectrum E central ( k ) = − t cos k , where k =2 πn/L z with n = 1 , ..., L z . Note that these are plane-waves of valley Gaussians, as can be seen from Fig. 2(b),which depicts the total density of the central band. No-tably, the bandwidth of the central band, 4¯ t , is muchsmaller than the gap to the bands of the first excitedstates √ t/ξ , as seen in Fig. 1(a). Therefore weak in-teraction and low temperatures will not mix it with theother bands.Turning on the repulsive interaction U , the effectivemodel of the central band enables us to describe the in-crease in compressibility using mean-field theory. Here, (cid:80) j n j +1 n j is approximated by (cid:80) j [ (cid:0) (cid:104) n j +1 (cid:105) + (cid:104) n j − (cid:105) (cid:1) n j −(cid:104) n j (cid:105)(cid:104) n j +1 (cid:105) − (cid:104) p j (cid:105) c † j +1 c j − (cid:104) p j (cid:105) ∗ c † j c j +1 + |(cid:104) p j (cid:105)| ], with (cid:104) n j (cid:105) as the background density, and (cid:104) p j (cid:105) = (cid:68) c † j c j +1 (cid:69) as thebackground exchange energy, both created by the occu-pied satellite bands below the central band. The constantterms do not contribute to ∆ ( N ), and will thereforebe ignored. The mean-field approximation adds effec-tive single-particle on-site potential and hopping, whichare modified according to the background density andexchange energy.We therefore turn to estimate (cid:104) n j (cid:105) and (cid:104) p j (cid:105) , and be-gin with solving the simplest Hamiltonian of Eq. (1)with U = (cid:15) = 0. This Hamiltonian describes a uni-form staggered potential ( − j λ cos φ . Its spectrum isgapped, unless the staggered potential is turned off at φ = π/
2. If the lower band is fully occupied, then themany-body density is also staggered, (cid:104) n j (cid:105)| (cid:15) =0 = 1 / − ( − j ¯ n ( λ cos φ/ t ), whereas the many-body exchange en-ergy is constant in space (cid:104) p j (cid:105)| (cid:15) =0 = ¯ p ( λ cos φ/ t ), where¯ n ( x ) = π − sign( x )K( − x − ) , (7)¯ p ( x ) = − π − | x | (cid:2) E( − x − ) − K( − x − ) (cid:3) , (8)and K( x ) and E( x ) are the complete elliptical integralsof the first and second kind, respectively [31].For (cid:15) (cid:54) = 0, the on-site potential corresponds locally to( − j , while λ cos φ varies slowly in space. Therefore, weexpect that the above expressions remain valid locallyand vary slowly in space, (cid:104) n j (cid:105) ≈ / − ( − j ¯ n [ λ cos(2 π(cid:15)j + φ ) / t ] , (9) (cid:104) p j (cid:105) ≈ ¯ p [ λ cos(2 π(cid:15)j + φ ) / t ] . (10)Fig. 2(b) depicts the background density (cid:104) n j (cid:105) obtainedboth numerically and analytically, according to Eq. (9),and they fit very well. It can be seen that (cid:104) n j (cid:105) followsthe cosine modulation. Therefore, between the valleys, (cid:104) n j (cid:105) ≈ / − ( − j ¯ n ( λ/ t ) cos(2 π(cid:15)j + φ ), to first approx-imation. The background exchange energy, (cid:104) p j (cid:105) , is ap-proximately uniform in space, and thus, (cid:104) p j (cid:105) ≈ ¯ p ( λ/ t ).Substituting these simplifications in the mean-field ap-proximation of H , we find that the background densityincreases the modulated on-site potential, and the ex-change energy enhances the hopping, H MF = L (cid:88) j =1 (cid:104) t eff c † j c j +1 + h.c + λ eff cos(2 πbj + φ ) n j + U n j (cid:105) , (11)where λ eff = λ + 2 U ¯ n ( λ/ t ) and t eff = t + U ¯ p ( λ/ t ).Like H ( U = 0), H MF has a central band of superlatticestates. Nevertheless, the width of the valley states, ξ ,and their hopping amplitude, ¯ t , are here determined by λ eff and t eff , rather than by λ and t . Although both λ eff and t eff increase with U , λ eff grows faster. Therefore, ξ = ξ ( t eff /λ eff ) decreases as a function of U , making theGaussians squeezed. Consequently, ¯ t also reduces, andthe central band becomes narrower, see Figs. 3(a)-(c).Intuitively, it is caused by the fact that the backgrounddensity follows the on-site potential, whereas the statesof the central bands are localized in its valleys. There-fore, the repulsion from background density squeezes theGaussians and reduces their overlap.In order to recover the enhanced compressibility in theinteracting case, one can diagonalize the effective nonin-teracting model of the central band H central [cf. Eq. (6)]with ¯ t ( t eff , λ eff ), and extract ∆ . Fig. 3(d) depicts∆ ( N, U ) that is obtained following this procedure.Clearly, it fits nicely to the one observed by DMRG.We note that the low-energy excitations of the satellitebands are also composed of valley states, and thereforethey also have decreasing compressibility, as implied byFig. 1(b). However, since the gap that separates themfrom the background states is much smaller, they willmix for much weaker interaction and lower temperatures.
FIG. 3.
Mean-field theory: (a)-(c) The effect of interac-tion between the central band and the background density,parametrized by the strength U , on: (a) the effective on-siteand hopping amplitudes, λ eff and t eff , for bare t = 1 and λ = 0 . ξ [cf. Eq. (5)]; and (c) the resulting hopping amplitude¯ t of the central band’s effective model [cf. Eq. (6)]. (d) Theinverse compressibility of the central band obtained by theeffective model of the central band (open circles) comparedto that obtained by DMRG (solid lines and dots, cf. Fig. 1).Increasing interaction corresponds to darker (green) shades. To summarize, we studied the Harper model of spinlessfermions at half-filling with nearest-neighbors repulsiveinteraction, for modulation frequency b = 1 / (cid:15) . Wefind that in the absence of interaction, a narrow band ap-pears at zero energy, which is separated by a gap from theother bands. This band is composed of states that are lo-calized within the superlattice of valleys of the potential.Conversely, the lower occupied bands form a backgrounddensity that follows the on-site potential. The strengthof the on-site potential that is experienced by the valleystates is increased by the repulsion from the background, λ eff > λ . As a result, these states become narrower, i.e.,their overlap diminishes, and their inverse compressibility∆ decreases. This unexpected effect may hint for an ex-planation for the above mentioned experiments [11, 12].Furthermore, it provides new insight on the interplay be-tween localization and e-e interaction. Last, our model isreadily implemented in existing technology of cold atomsin optical lattices [5, 6]. Notably, using hard-core bosons,rather than spinless fermions, will increase the effect,since now the effective hopping is reduced by interaction t eff < t .We would like to thank E. Berg, S. Huber, and R. Chi-tra for useful discussions. Financial support from theIsrael Science Foundation (Grant 686/10), the US-IsraelBinational Science Foundation, the Minerva foundationand the Swiss National Foundation is gratefully acknowl-edged. [1] L. Li, C. Richter, S. Paetel, T. Kopp, J. Mannhart, andR. C. Ashoori, Science , 885 (2011).[2] V. Tinkl, M. Breitschaft, C. Richter, and J. Mannhart,Phys. Rev. B , 075116 (2012).[3] Y. Alhassid, Rev. Mod. Phys. , 895 (2000).[4] K. K. Gomes, W. Mar, W. Ko, F. Guinea, and H. C.Manoharan, Nature , 306 (2012).[5] I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. , 885 (2008).[6] G. Roati et al., Nature , 895 (2008).[7] M. Aidelsburger, M. Atala, M. Lohse, J. T. Barreiro,B. Paredes, and I. Bloch, Phys. Rev. Lett. , 185301(2013).[8] H. Miyake, G. A. Siviloglou, C. J. Kennedy, W. C. Bur-ton, and W. Ketterle, Phys. Rev. Lett. , 185302(2013).[9] T. Roscilde, New J. Phys. , 023019 (2009).[10] T. Giamarchi, Quantum Physics in One Dimension (Ox-ford, 2003).[11] R. C. Ashoori, H. L. Stormer, J. S. Weiner, L. N. Pfeiffer,S. J. Pearton, K. W. Baldwin, and K. W. West, Phys.Rev. Lett. , 3088 (1992).[12] N. B. Zhitenev, R. C. Ashoori, L. N. Pfeiffer, and K. W.West, Phys. Rev. Lett. , 2308 (1997).[13] P. G. Harper, Proc. Phys. Soc. London A , 874 (1955).[14] S. Aubry and G. Andr´e, Ann. Isr. Phys. Soc. , 133(1980).[15] For a review see H. Hiramoto and M. Kohmoto, Int. J.Mod. Phys. B , 281 (1992).[16] S. Y. Jitomirskaya, Annals of Mathematics , 1159(1999).[17] Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti,N. Davidson, and Y. Silberberg, Phys. Rev. Lett. ,013901 (2009).[18] J. Chab´e, G. Lemari´e, B. Gr´emaud, D. Delande, P. Szrift-giser, and J. C. Garreau, Phys. Rev. Lett. , 255702(2008).[19] G. Modugno, Rep. Prog. Phys. , 102401 (2010).[20] J. Vidal, D. Mouhanna, and T. Giamarchi, Phys. Rev.Lett. , 3908 (1999).[21] J. Vidal, D. Mouhanna, and T. Giamarchi, Phys. Rev. B , 014201 (2001).[22] C. Schuster, R. A. R¨omer, and M. Schreiber, Phys. Rev.B , 115114 (2002).[23] S. Iyer, V. Oganesyan, G. Refael, and D. A. Huse, Phys.Rev. B , 134202 (2013).[24] Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zil-berberg, Phys. Rev. Lett. , 106402 (2012).[25] Y. E. Kraus and O. Zilberberg, Phys. Rev. Lett. ,116404 (2012).[26] M. Verbin, O. Zilberberg, Y. E. Kraus, Y. Lahini, andY. Silberberg, Phys. Rev. Lett. , 076403 (2013).[27] Y. E. Kraus, Z. Ringel, and O. Zilberberg,arXiv:1308.2378 (2013).[28] S. R. White, Phys. Rev. Lett. , 2863 (1992).[29] S. R. White, Phys. Rev. B , 10345 (1993).[30] The subgap states that appear near the bands are topo-logical boundary states [24].[31] For the full derivation, see Supplemental Material. SUPPLEMENTAL MATERIAL
I. SPECTRUM OF VALLEY HAMILTONIAN
In the main text, we presented the approximatedHamiltonian of the noninteracting case within a val-ley of the potential, and its Fourier transform, H valley [cf. Eq. (3) of the main text]. By rotating it in thesublattice space with the rotation operator ˆ T = (1 + is z σ x ) / √
2, and introducing the ladder operator a k = − ( k − π/ ξ/ √ i (ˆ p k − l z ) / √ ξ , we turned H valley intosupersymmetric form [cf. Eq. (4) of the main text]. Herewe derive its energy spectrum and eigenstates.A supersymmetric Hamiltonian has a zero-energyeigenstate. We find it by solving the differential equa-tion a k ϕ ( k ) = 0, which gives ϕ ( k ) = (cid:112) L k ( ξ /π ) / e − il z k e − ( k − π/ ξ / , (I.1)where L k = 2 π/L . For higher energy eigenstates, wedefine the functions ϕ n ( k ) = ( a † k ) n ϕ ( k ) / √ n !, which sat-isfy the relations a k ϕ n ( k ) = √ nϕ n − ( k ) and a † k ϕ n ( k ) = √ n + 1 ϕ n +1 ( k ). Basically, these functions are the eigen-states of the harmonic oscillator, shifted by π/ e il z l .The corresponding eigenstates of H valley are¯ ϕ † n, ± = (cid:90) π dk √ L k ( ˆ T ψ k ) † (cid:18) ϕ n ( k ) ± ϕ n − ( k ) (cid:19) = (cid:90) π dk L k ψ † k (cid:18) ϕ n ( k ) ± is z ϕ n − ( k ) is z ϕ n ( k ) ± ϕ n − ( k ) (cid:19) . (I.2)Note that for n = 0 there is only one solution, with ϕ − ( k ) ≡
0, and we multiply this expression by √ E valley n, ± = ± t √ n/ξ , as mentionedin the main text. Notably, in its supersymmetric form, H valley anticommutes with σ z , and thus, the states comeas particle-hole pairs, ¯ ϕ † n, − = σ z ¯ ϕ † n, + . The only unpairedstate is the zero-energy state, ¯ ϕ , which is also protectedat zero energy, as long as the anticommutation holds.We can see that all the eigenstates have the same phasefactor e − il z k and are confined by the same Gaussian e − ( k − π/ ξ / . Therefore, in real space, they are alsoconfined by a Gaussian e − ( j − l z ) / ξ around the node l z ,and are accompanied by the phase factor e ijπ/ = i j .We are mostly interested in the wave function of the zero-energy state,¯ ϕ † = (cid:90) π dk √ L k ( c † e k , c † o k ) (cid:18) ϕ ( k ) is z ϕ ( k ) (cid:19) (I.3)= L/ (cid:88) j =1 (cid:90) π dk √ πL k [ ϕ ( k ) e ik j c † j + is z ϕ ( k ) e ik (2 j − c † j − ]= ( πξ ) − / e − il z π/ (cid:80) L/ j =1 ( − j × (cid:16) e − (2 j − l z ) / (2 ξ ) c † j − s z e − (2 j − − l z ) / (2 ξ ) c † j − (cid:17) . By defining | l z (cid:105) ≡ ¯ ϕ † | vacuum (cid:105) and S j = √ jπ/ − π/
4) = . . . , − , − , , , − , − , . . . ,and omitting the global phase factor, we obtain Eq. (5)of the main text. II. HOPPING TERM OF CENTRAL-BANDHAMILTONIAN
The central band of the noninteracting spectrum is ourmain interest. This band is composed of multiple single-valley states | l z (cid:105) [cf. Eq. (5) of the main text], whichoverlap and thus hop. In order to get an effective Hamil-tonian for this band, we project the full noninteractingHamiltonian H ( U = 0) [cf. Eq. (1) of the main text] ontothe space of the | l z (cid:105) states. In the main text we have seenthat the only relevant matrix elements of the projectionare (cid:104) l z | H ( U = 0) | l z ± (cid:105) , and here we evaluate them.Recall that l z ± = l z ± / | (cid:15) | , and accordingly we cansubstitute cos(2 π(cid:15)j + φ ) = ± s z cos[2 π(cid:15) ( j − l z ∓ / | (cid:15) | )].We can now evaluate (cid:104) l z | H ( U = 0) | l z ± (cid:105) = (cid:80) Lj =1 (cid:104) l z | tc † j c j +1 + h.c | l z ± (cid:105)± s z λ cos[2 π(cid:15) ( j − l z ∓ / (cid:15) )] (cid:104) l z | c † j c j | l z ± (cid:105) = s z √ πξ L (cid:88) j =1 (cid:20) ± λ e − [( n − l z ) +( n − l z ∓ / | (cid:15) | ) ] / (2 ξ ) × ( e i π(cid:15) ( n − l z ∓ / | (cid:15) | ) + e − i π(cid:15) ( n − l z ∓ / | (cid:15) | ) )+ te − [( n − l z ) +( n − − l z ∓ / | (cid:15) | ) ] / (2 ξ ) − te − [( n − l z ) +( n +1 − l z ∓ / | (cid:15) | ) ] / (2 ξ ) (cid:105) (I.4)where we used the relation S j ± = ± ( − j S j . Using alsothe fact that ξ (cid:29)
1, we shift the summation, and obtain (cid:104) l z | H ( U = 0) | l z ± (cid:105) ≈ ± s z e − / (4 ξ(cid:15) ) × (I.5) λ e − ( π(cid:15)ξ ) · √ πξ L (cid:88) j =1 (cid:0) e − ( j − iπ(cid:15)ξ ) /ξ + e − ( j + iπ(cid:15)ξ ) /ξ (cid:1) + te − / (4 ξ ) (cid:0) e − / (4 ξ | (cid:15) | ) − e / (4 ξ | (cid:15) | ) (cid:1) · √ πξ L (cid:88) j =1 e − j /ξ . Performing the Gaussian sums, we obtain Eq. (6) of themain text, with the corresponding ¯ t . III. SOLVING H FOR b = 1 / AND U = 0 In the mean-field theory that we use in the main text,we present the many-body density, (cid:104) n j (cid:105)| (cid:15) =0 , and ex-change energy, (cid:104) p j (cid:105)| (cid:15) =0 , of the lower band of the simplenoninteracting Hamiltonian with b = 1 /
2, i.e. U = (cid:15) = 0.Here we derive the expression for (cid:104) n j (cid:105)| (cid:15) =0 and (cid:104) p j (cid:105)| (cid:15) =0 .For b = 1 /
2, our Hamiltonian H [cf. Eq. (1) of themain text] becomes H (cid:15) =0 = (cid:80) Lj =1 (cid:104) tc † j c j +1 + h.c + λ cos φ ( − j n j (cid:105) = (cid:90) π dkL k ψ † k [2 t cos k σ x + λ cos φ σ z ] ψ k , (I.6)where ψ k = ( c e k , c o k ) T is the sublattice pseudospinor,the same as in the main text [cf. Eq.(3)]. The energyspectrum of H (cid:15) =0 is composed of two bands, E (cid:15) =0 k, ± = ± (cid:112) t cos k + λ cos φ . The corresponding eigenstatesare χ † k, ± = (cid:112) L/ c † e k , c † o k ) (cid:18) χ e k, ± χ o k, ± (cid:19) (I.7)= L (cid:88) j =1 (cid:0) χ e k, ± e ik j c † j + χ o k, ± e ik (2 j − c † j − (cid:1) , (I.8)where (cid:18) χ e k, ± χ o k, ± (cid:19) = 1 (cid:113) E (cid:15) =0 k, ± ( E (cid:15) =0 k, ± − λ cos φ ) (cid:18) t cos kE (cid:15) =0 k, ± − λ cos φ (cid:19) The density of an eigenstate is (cid:104) χ k, ± | n j | χ k, ± (cid:105) = χ k, ± + χ k, ± − j χ k, ± − χ k, ±
2= 12 ∓
12 sign( λ cos φ ) (cid:112) λ cos φ/ t ) − cos k , (I.9)and its exchange energy is (cid:104) χ k, ± | c † j +1 c j | χ k, ± (cid:105) = e − ik χ e k, ± χ o k, ± (I.10)= ± t | λ cos φ | e − ik cos k (cid:112) λ cos φ/ t ) − cos k . If the lower band is fully occupied, then the many-bodydensity is given by (cid:104) n j (cid:105)| (cid:15) =0 = (cid:90) π/ − π/ dkπ (cid:104) χ k, − | n j | χ k, − (cid:105) = 1 / − ( − j ¯ n ( λ cos φ/ t ) , (I.11) where¯ n ( x ) = (cid:90) π/ − π/ dk π sign( x ) (cid:113) − sin k ) /x = xπ √ x (cid:90) π/ dk (cid:113) − (1 + x ) − sin k = xπ √ x K (cid:18)
11 + x (cid:19) , (I.12)with K( x ) as the complete elliptical integral of the firstkind. If one uses a definition of K( x ) in which x is notlimited to the interval (0 , n ( x ) can be furthersimplified to yield Eq. (7) of the main text. Similarly,the many-body exchange energy is (cid:104) p j (cid:105)| (cid:15) =0 = (cid:90) π/ − π/ dkπ (cid:104) χ k, − | c † j +1 c j | χ k, − (cid:105) = ¯ p ( λ cos φ/ t ) , (I.13)with¯ p ( x ) = − π √ x (cid:90) π/ − π/ dk cos k − i sin k cos k (cid:113) − (1 + x ) − sin k The imaginary part of the integral vanishes due its anti-symmetry in the interval. Therefore,¯ p ( x ) = 1 π √ x (cid:90) π/ dk x (cid:113) − (1 + x ) − sin k − (1 + x ) (cid:113) − (1 + x ) − sin k (cid:21) (I.14)= x π √ x K (cid:18)
11 + x (cid:19) − √ x π E (cid:18)
11 + x (cid:19) . Here E( x ) is the complete elliptical integrals of the sec-ond kind. Again, for unlimited x , ¯ p ( xx