PRL 111, 196401 (2013)

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Topological Edge States and Fractional Quantum Hall Effect from Umklapp Scattering Jelena Klinovaja and Daniel Loss Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland (Received 25 February 2013; published 4 November 2013) We study anisotropic lattice strips in the presence of a magnetic field in the quantum Hall effect regime. At specific magnetic fields, causing resonant umklapp scattering, the system is gapped in the bulk and supports chiral edge states in close analogy to topological insulators. In electron gases with stripes, these gaps result in plateaus for the Hall conductivity exactly at the known fillings n=m (both positive integers and m odd) for the integer and fractional quantum Hall effect. For double strips, we find topological phase transitions with phases that support midgap edge states with flat dispersion. The topological effects predicted here could be tested directly in optical lattices. DOI: 10.1103/PhysRevLett.111.196401

PACS numbers: 71.10.Fd, 05.30.Pr, 71.10.Pm, 73.43.f

Condensed matter systems with topological properties have attracted wide attention over the years [1–5]. For example, the integer and fractional quantum Hall effects (IQHE and FQHE, respectively) [6,7] find their origin in the topology of the system [8–19]. Similarly, band insulators with topological properties have become of central interest recently [2,3,20], as well as exotic topological states like fractionally charged fermions [21–29] or Majorana fermions [30–40]. Here, we study two-dimensional (2D) strips in magnetic fields, both analytically and numerically, modeled by an anisotropic tight-binding lattice. We identify a striking mechanism by which the magnetic field induces resonant umklapp scattering (across Brillouin zones) that opens a gap in the bulk spectrum and results in chiral edge states in analogy to topological insulators. Quite remarkably, at quarter filling the resonant scattering occurs at well-known filling factors for the IQHE [6] and FQHE [7]  ¼ n=m, where n and m are positive integers and m is odd. This mechanism could shed new light on the QHE for 2D electron gases as well, where the formation, e.g., of a periodic structure (energetically favored also by a Peierls transition) might support the periodic structure needed for the umklapp scattering. We further consider a double strip of spinless fermions and a single strip with spinful fermions. In both models, we discover two topological phase transitions accompanied by a closing and reopening of the bulk gap and, as a result, three distinct phases. The trivial phase is without edge states. The first topological phase is similar to the one discussed above and carries two propagating chiral modes at each edge for  ¼ 1. The second topological phase has only one state at each edge. Quite remarkably, the edge-state dispersion is flat throughout the Brillouin zone, making this phase an attractive playground for studying interaction effects. We note that in contrast to flat band Chern insulators [41–43], which also need fine-tuned hopping beyond the nearest neighbor, it is only the edge but not the bulk states that are flat in our model. As a remarkable 0031-9007=13=111(19)=196401(5)

consequence, such edge states are fully gapped away from the bulk states. Given that our models are based on singleparticle Hamiltonians without interaction terms, it should be possible to implement them in optical lattices [44]. Anisotropic tight-binding model.—We consider a 2D tight-binding model of a strip that is of width W in the x and extended in the y direction; see Fig. 1(a). The unit cell is composed of two lattice sites ( ¼ 1) along y that are distinguished by two hopping amplitudes ty1 and ty2 . Every site is labeled by three indices n, m, and , where n (m) denotes the position of the unit cell along the x (y) axis. The hopping along x is described by X y ðcnþ1;m; cn;m; þ H:c:Þ; (1) Hx ¼ tx n;m;

where tx is the hopping amplitude in the x direction and cn;m; the annihilation operator acting on a spinless



FIG. 1 (color online). (a) Strip: two-dimensional lattice of width W in the x direction with the unit cell defined by the lattice constants ax and ay . The hopping amplitudes in the y direction, ty1 and ty2 , carry the phase , arising from a perpendicular magnetic field B, and we assume tx  ty1 , ty2 . (b) Doubly degenerate spectrum of Hx [see Eq. (1)] for the rows with  ¼ 1 (upper blue line) and with  ¼ 1 (lower green line). The hoppings ty1 (dashed line) and ty2 (dotted line) induce resonant scattering between right (R ) and left (L ) movers, which open gaps (not shown) at the Fermi wave vectors kF defined by the chemical potential .


Ó 2013 American Physical Society

PRL 111, 196401 (2013)


fermion at site (n, m, ), and the sum runs over all sites. The hopping along y is described by X Hy ¼ ðty1 ein cyn;m;1 cn;m;1 þty2 ein cyn;mþ1;1 cn;m;1 þH:c:Þ: n;m

(2) Without loss of generality, we consider ty2  ty1  0. The phase  is generated by a uniform magnetic field B applied in the perpendicular z direction; see Fig. 1. We choose the corresponding vector potential A to be along the y axis, A ¼ ðBxÞey , yielding the phase  ¼ eBax ay [email protected] Here, ax;y are the corresponding lattice constants. We note that if ax;y are kept fixed and independent of any parameters such as B or density (but see below), we recover the Hofstadter model in the uniform limit ty1 ¼ ty2 [8]. Chiral edge states.—Taking into account translational invariance of the system in the y direction, we introduce the momentum ky via Fourier transformation [45]. The Hamiltonians become diagonal in ky , P i.e., Hx ¼ tx n;ky ; ðcynþ1;ky ; cn;ky ; þ H:c:Þ and Hy ¼ P in þ ty2 eiðnky ay Þ Þcyn;ky ;1 cn;ky ;1 þ H:c:. Thus, n;ky ½ðty1 e the eigenfunctions of H ¼ Hx þ Hy factorize as eiky y c ky ðxÞ, where we focus now on c ky ðxÞ and treat ky as a parameter. Assuming for the moment periodic boundary conditions also in the x direction, we introduce a momentum kx [45]. Immediately, the well-known spectrum of Hx follows:  ¼ 2tx cosðkx ax Þ, which is twofold degenerate in . The chemical potential  is fixed such that the Fermi wave vector kF is connected to the phase by  ¼ 2kF ax . Next, we allow for hopping along y as a small perturbation to the x hopping, i.e., tx  ty1 , ty2 ; see Fig. 1(b). To obtain analytical solutions, it is most convenient to go to the continuum description [28,46]. The annihilation operator ðxÞ close to the Fermi level can be represented in terms of slowly varying right [R ðxÞ] and left [L ðxÞ] movers: P ðxÞ ¼  R ðxÞeikF x þ L ðxÞeikF x . The corresponding Hamiltonian density H can be rewritten in terms of the Pauli matrices i (i ), acting on the right-left mover (lattice) subspace [45],  ¼ ðR1 ; L1 ; R1 ; L1 Þ, as ^ 3þ H ¼ @F k

ty1 ð  þ 2 2 Þ 2 1 1

ty2 ½ð1 1  2 2 Þ cosðky ay Þ 2 þ ð2 1 þ 1 2 Þ sinðky ay Þ:



Here, @k^ ¼ [email protected]@x is the momentum operator with eigenvalues k taken from the corresponding Fermi points kF , and F ¼ 2ðtx [email protected]Þax sinðkF ax Þ is the Fermi velocity. The spectrum with periodic boundary conditions in the x and y qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi directions is given by l; ¼  ð@F kÞ2 þ t2yl , where l ¼ 1; 2. This mechanism of opening a gap by oscillatory terms

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causing resonant scattering between the Fermi points is similar to a Peierls transition [47]. Next, we turn to a strip of finite width W; see Fig. 1. We note that the bulk spectrum l; is fully gapped, so states localized at the edges can potentially exist. To explore this possibility we consider a semi-infinite nanowire (x  0) and follow the method developed in Refs. [28,46], assuming that the localization length of bound states  is much smaller than W. This allows us to impose vanishing boundary conditions only at x ¼ 0: c ky ðxÞjx¼0 ð c 1 ; c 1 Þjx¼0 ¼0. This boundary condition is fulfilled only at one energy inside the gap jEj < ty1 , ty1 ty2 sinðky ay Þ Eðky Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; t2y1 þ t2y2  2ty1 ty2 cosðky ay Þ


if the following condition is satisfied: ty1 > ty2 cosðky ay Þ:


The edge states exist for momenta ky 2 ðk ; kþ Þ, where k ay ¼  ½ =2 þ arcsinðty1 =ty2 Þ. An edge state touches a boundary of the gap at k and afterwards disappears in the bulk spectrum of the delocalized states; see Fig. 2. The only regime in which the edge state exhibits all momenta corresponds to the uniform strip with ty1 ¼ ty2 . The localization length  is determined by  ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @F = t2y1  E2 , with the wave function given in Ref. [45]. The edge state gets delocalized if its energy is close to the boundary of the gap, so that  becomes comparable to W. Similarly, we can search for the solution decaying to the right, x  0, and obtain Eq. (4) with reversed sign, Eðky Þ ! Eðky Þ. We have confirmed the above results by diagonalizing the tight-binding Hamiltonian H (in ky representation) numerically; see Fig. 2. The spectrum Eðky Þ of the edge states localized along x and propagating along y shows that at any fixed energy inside the gap there can be only one edge state at a given edge; see Fig. 2. Moreover, the edge states are chiral, as can be seen from the velocity  ¼ @[email protected] , which is negative (positive) for the left (right) edge state. This means that transport along a given edge of the strip can occur only in one direction determined by the direction of the B field; see Fig. 1. Quite remarkably, the obtained spectrum of edge states is of the same form as for topological insulators [2,5,48] with a single Dirac cone consisting of two crossing nondegenerate subgap modes. Because of the macroscopic separation of opposite edges,   W, these modes are protected from getting scattered into each other by impurities, phonons, or interaction effects, so that the Dirac cone cannot be eliminated by perturbations that are local and smaller than the gap. Thus, the edge states are topologically stable.



PRL 111, 196401 (2013)







FIG. 2 (color online). Spectrum Eðky Þ of the left edge state (red line or dots) propagating along y for a strip (ty1 =tx ¼ 0:02, ty2 =tx ¼ 0:1) of width W=ax ¼ 801 and with phase  ¼ =2, obtained (a) analytically [see Eq. (4)] and (b) numerically [see Eqs. (1) and (2)]. For k < ky < kþ , there exists one edge state at each edge. The left (red dots) and the right (blue dots) edge states are chiral and propagate in opposite y directions. For each ky [(c) ky ay ¼ , (d) ky ay ¼ 13 =12] there is one left (red dots) and one right (blue dots) edge state if Eq. (5) is satisfied. Here, ðNÞ corresponds to the Nth energy level. The probability density j c  j2 (e) [(f)] of the left [right] localized state decays exponentially in agreement with the analytical result, Eq. (4) in Ref. [45].

Umklapp scattering.—We note that the system considered here is equivalent to a 2D system in the QHE regime. The above choice of magnetic field corresponds to the IQHE with filling factor  ¼ 1, which is in agreement with one chiral mode at each edge. To explore the possibility of inducing quantum Hall physics at other pffiffiffi filling factors, we fix the chemical potential  ¼  2tx and change the B field. Above, the phase , generated by the magnetic field, was equal to =2 for kF ¼ =4ax . However, this is not the only choice of phase leading to the opening of a gap g at the Fermi level. Because of the periodicity of the spectrum, resonant scattering between branches of  occurs also via umklapp scattering between different Brillouin zones, with a phase  ¼ ðp =2nÞ þ 2 q, where q is an integer, n a positive integer (corresponding to nth-order perturbation), and p a positive odd integer with p < 2n and coprime to n (see the Supplemental Material for more details [45]). As a result, the Fermi level lies in the bulk gap for the filling factors [49]  ¼ n=ð4qn  pÞ, which can be rewritten as  ¼ n=m, where m > 0 is an odd integer. The size of the gap can be estimated as g / tyl ðtyl =tx Þðn1Þ (assuming for simplicity ty1 ¼ ty2 ). Finally, we remark that we checked

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numerically that the gap g never closes for any finite ratio of tx and tyl larger or smaller than one. FQHE in 2D electron gas.—We conjecture that the same mechanism of resonant umklapp scattering can also lead to the integer or fractional QHE in 2D electron gases. At high magnetic fields, interaction effects get strongly enhanced and electrons tend to order themselves into periodic structures [16,50–56]. In particular, we assume the formation of stripes that are aligned along x and periodically repeated in y. While particles can hop between stripes, they move now continuously inside them with quadratic dispersion. Thus, the perturbative solutions found above in terms of right and left movers still apply. In addition, we assume that the interaction generates a charge-density wave at wave vector K inside the stripe, providing an effective periodic potential in x, which will lead to gaps. Thus, K becomes the period of the Brillouin zone, and, at 1=4 filling of the lowest subband, we have K ¼ 8kF [57]. Again, the B field leads to a gap at kF only if it results in phases commensurable to kF , i.e., eBay [email protected] ¼ 2kF ðp=nÞ þ qK, which is equivalent to  ¼ n=ð4qn  pÞ  n=m. In this regime, there is an additional energy gain due to a Peierls transition, favoring even more a formation of periodic structures with gaps. Moreover, from this mean field scenario it follows that the IQHE is more stable against disorder than the FQHE, since the latter requires umklapp scattering through higher Brillouin zones. The gap and the edge states can be tested in transport experiments. For example, the Hall conductance H exhibits plateaus on the classical dependence curve H / 1=B, if the Fermi level lies in the gap. This can be shown by using the   is Streda formula [11] H ¼ ecð@[email protected]Þ  , where n the bulk particle density which is uniquely determined by the magnetic field via the relation eBay [email protected] ¼ 2kF (for this, it is crucial that K depends on kF ). If  lies in the gap opened by the umklapp scattering, the change in the density  due to a change in the magnetic field, dB, is for fixed , dn, given by dn ¼ ðdkF = ay =2Þ ¼ ðe=hcÞdB. Hence, the conductance assumes the FQHE plateaus, H ¼ e2 =hc, with  ¼ n=m and independent of any lattice parameters. The width of the plateaus is determined by the gap size g / tyl ðtyl =Þðn1Þ . Finally, the distance between stripes  can be estimated as ay =2 ¼ kF = n. Double strip.—Now we consider a double strip, consisting of two coupled strips for spinless fermions; see Fig. 3.

FIG. 3 (color online). Double strip for spinless fermions. The intrastrip couplings are the same as in Fig. 1. The interstrip coupling along z is described by the hopping amplitude tz .



PRL 111, 196401 (2013) (a)


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FIG. 4 (color online). Spectrum of a double strip obtained by numerical diagonalization of the tight-binding Hamiltonian H2 ¼ Hx þ Hy þ Hz for the same parameters as in Fig. 2. (a) If jtz j < ty1 [tz =tx ¼ 0:01], there are four edge states at any energy within the gap; two of them localized at the left and two at the right edge. (b) If ty1 < jtz j < ty2 [tz =tx ¼ 0:05], there is one zero-energy edge state (i.e., with flat dispersion) at each edge. (c) If jtz j > ty2 [tz =tx ¼ 1:5], there are no edge states in the gap.

This system is equivalent to a single strip but for spinful fermions. Below, we focus on the double strip, but we note that one can identify the upper (lower) strip with the spin-up (-down) state labeled by ¼ 1 ( ¼ 1). The chemical potentials  are opposite for the two strips, pffiffiffi 1 ¼ 1 ¼ 2tx , and are chosen such that the system is at half filling. For the spinful strip the role of  is played by the Zeeman term,  ¼ gB B, arising from the magnetic field B along z. Here, g is the g factor, and B is the Bohr magneton. The interstrip hopping amplitude tz is also accompanied by the phase z arising from a uniform magnetic field B2 applied along y,

c LE¼0 ¼ ðfðxÞ; if ðxÞ; ið1Þn fðxÞ; ð1Þnþ1 f ðxÞÞ; fðxÞ ¼ eiky ay =2 eðk2 þikF1 Þx  cosðky ay =2Þeðk1 ikF1 Þx þ i sinðky ay =2Þeðk1þ ikF1 Þx :



The basis ( c 1;1 , c 1;1 , c 1;1  , c 1;  1 ) is composed of wave functions c ; defined at the -unit lattice site of the

strip. The smallest wave vectors kl; ¼ jtl  tz [email protected]F determine the localization length of the edge state. We note that the probability densities j c ; ðxÞj2 are uniform inside the unit cell. If jtz j < ty1 , there are two edge states at each edge for E inside the gap; see Fig. 4. These states, propagating in y, have a momentum ky determined by E. This case is similar to one strip with spinless particles discussed above. We note that the edge states found here are the higherdimensional extensions of the end bound states found in one-dimensional nanowires [28,58] and ladders [29]. For jtz j > ty2 , there are no edge states; see Fig. 4. Finally, the most interesting regime here is ty1 < jtz j < ty2 , where there is one zero-energy edge state at each edge; see Fig. 4. Such states with flat dispersion are expected to be strongly affected by interactions. Conclusions.—We have studied topological regimes of strips with modulated hopping amplitudes in the presence of magnetic fields. We found topological regimes with chiral edge states at filling factors that correspond to integer and fractional QHE regimes. We showed that double strips sustain topological phases with midgap edge states with flat dispersion. This work is supported by the Swiss NSF, NCCR Nanoscience, and NCCR QSIT.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The resulting spectrum is l;;p ¼ ð@F kÞ2 þðtyl þptz Þ2 , with p ¼ 1. We note that the gap vanishes if jtz j ¼ ty1 or jtz j ¼ ty2 . The closing and reopening of a gap often signals a topological phase transition. Indeed, imposing vanishing boundary conditions at the edges, we find that there are two edge states (one at each edge) at zero energy E ¼ 0, if the following topological criterion is satisfied: ty1 < jtz j < ty2 ; see Fig. 4. The wave function of the left edge state for tz > 0 is given by (with x ¼ nax )

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Hz ¼


tz ein  cyn;m;; cn;m;;  : z



The amplitude of B2 is chosen so that z ¼ ð[email protected]ÞB2 ax az ¼ . This amounts to applying a total field Btot ¼ B þ B2 in the yz plane. Moreover, the same Hz is generated in the spinful case by a B2 field applied along y with an amplitude that oscillates in space along x with period 2ax or, alternatively, by Rashba spin orbit interaction [47]. Again, we search for wave functions in terms of right and left mover fields defined around two Fermi points, kF1 ¼ =4ax (upper strip) and kF1 ¼ 3 =4ax (lower strip). The linearized Hamiltonian density for this extended model in terms of the Pauli matrices i acting on the upper or lower strip subspace is given by [see Eq. (3)] H 2 ¼ H ð2 ! 2 3 Þ þ tz 1 1 :


PRL 111, 196401 (2013)


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Topological edge states and fractional quantum Hall effect from umklapp scattering.

We study anisotropic lattice strips in the presence of a magnetic field in the quantum Hall effect regime. At specific magnetic fields, causing resona...
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