Fractional boundary charges with quantized slopes in interacting one- and two-dimensional systems
Katharina Laubscher, Clara S. Weber, Dante M. Kennes, Mikhail Pletyukhov, Herbert Schoeller, Daniel Loss, Jelena Klinovaja
FFractional boundary charges with quantized slopes in interactingone- and two-dimensional systems
Katharina Laubscher, Clara S. Weber, Dante M. Kennes,
2, 3
MikhailPletyukhov, Herbert Schoeller, Daniel Loss, and Jelena Klinovaja Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland Institut f¨ur Theorie der Statistischen Physik, RWTH Aachen, 52056 Aachen,Germany and JARA - Fundamentals of Future Information Technology Max Planck Institute for the Structure and Dynamics of Matter,Center for Free Electron Laser Science, 22761 Hamburg, Germany (Dated: January 26, 2021)We study fractional boundary charges (FBCs) for two classes of strongly interacting systems.First, we study strongly interacting nanowires subjected to a periodic potential with a period thatis a rational fraction of the Fermi wavelength. For sufficiently strong interactions, the periodicpotential leads to the opening of a charge density wave gap at the Fermi level. The FBC thendepends linearly on the phase offset of the potential with a quantized slope determined by theperiod. Furthermore, different possible values for the FBC at a fixed phase offset label differentdegenerate ground states of the system that cannot be connected adiabatically. Next, we turn tothe fractional quantum Hall effect (FQHE) at odd filling factors ν = 1 / (2 l + 1), where l is aninteger. For a Corbino disk threaded by an external flux, we find that the FBC depends linearly onthe flux with a quantized slope that is determined by the filling factor. Again, the FBC has 2 l + 1different branches that cannot be connected adiabatically, reflecting the (2 l + 1)-fold degeneracyof the ground state. These results allow for several promising and strikingly simple ways to probestrongly interacting phases via boundary charge measurements. Introduction.
The emergence of fractional chargesin topologically nontrivial systems is a recurring themein modern condensed matter physics that has beendiscussed in several different contexts. In the frac-tional quantum Hall effect (FQHE), for example, strongelectron-electron interactions lead to the emergence ofexotic quasiparticles carrying only a fraction of the elec-tronic charge e . Fractional charges were, however,also discussed in noninteracting models like the Jackiw-Rebbi and Su-Schrieffer-Heeger models as well asextensions thereof.
Here, domain walls between topo-logically nonequivalent gapped phases bind well-definedfractional charges that may be quantized due to sym-metry constraints. Via a similar mechanism, fractionalcharges can also accumulate at open boundaries of afinite insulating system. Importantly, while the possi-ble presence of edge states influences the total bound-ary charge only by an integer number, the fractional part of the boundary charge contains contributions fromall extended states and is directly related to bulk prop-erties via the Zak-Berry phase.
Fractional bound-ary charges (FBCs) of this type have been studied ina large variety of systems, including different types ofone-dimensional (1D) models, topological crystallineinsulators, higher-order topological insulators, and the integer quantum Hall effect (IQHE). The FBCsin these examples were found to display various inter-esting features such as quantization in the presence ofsymmetries, universal dependencies on certain systemparameters, or a direct relation to the Hall con-ductance in the case of the IQHE. This makes theFBC a quantity of high interest that is furthermoreaccessible to experiments in a rather straightforward
FIG. 1. Corbino disk in the FQHE regime threaded by an ex-ternal flux Φ. The FBC is measured in the red region, whichextends into the bulk on the order of a few edge state local-ization lengths ξ . In the presence of a constriction, indicatedby the dashed line, tunneling of fractional charges betweenthe chiral edge states (blue lines) is allowed. way using, e.g., scanning tunneling microscopy (STM)techniques. While previous studies were mainly concerned withnoninteracting systems, the aim of this work is to studyFBCs in gapped phases that can only arise in the pres-ence of strong electron-electron interactions. First, weconsider a 1D nanowire with a periodic potential of theform cos(2 mk F x + α ), where k F is the Fermi momen-tum, m an integer, and α a phase offset. For m = 1, itis well-known that a charge density wave (CDW) gap isopened at the Fermi level with an FBC that dependslinearly on α with a quantized slope 1 / π . In thepresence of strong interactions, additional gaps can beopened for m >
1. We show that in this case the FBC de- a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n pends again linearly on α with a fractional slope 1 / πm .Perhaps even more interestingly, we find that there arenow m degenerate ground states labeled by m differentbranches of the FBC that cannot be connected underadiabatic evolution of α .In the second part, we extend our considerations toa two-dimensional electron gas (2DEG) in the FQHEregime at odd filling factors ν = 1 / (2 l + 1), where l isan integer. For a Corbino disk threaded by an externalflux Φ as shown in Fig. 1, we find that the FBC dependslinearly on Φ with a slope determined by the filling fac-tor. From this the standard Hall conductance follows.Further, the FBC has 2 l + 1 distinct branches labeling2 l + 1 degenerate ground states that cannot be adiabat-ically connected unless fractional charges are allowed totunnel between the two boundaries due to, e.g., a con-striction, see again Fig. 1. We outline how these resultsopen up a strikingly simple way to probe strongly inter-acting systems via boundary charge measurements. FBC in one dimension.
We first study the FBC ina 1D nanowire of spinless electrons subjected to a spa-tially modulated potential V m ( x ) = 2 V m cos(2 mk F x + α ).Here, k F is the Fermi momentum, 2 V m > α a phase offset, and m an integer. The single-particle Hamiltonian reads H = Z dx Ψ † ( x ) (cid:20) − ~ ∂ x m ∗ + V m ( x ) (cid:21) Ψ( x ) , (1)where Ψ † ( x ) [Ψ( x )] creates [annihilates] a spinless elec-tron of mass m ∗ at the position x . Without interactions,the onsite potential opens a CDW gap at the Fermi levelonly for m = 1. To study the more general case of aninteracting system, we linearize the spectrum around theFermi points and write Ψ( x ) = R ( x ) e ik F x + L ( x ) e − ik F x ,where R ( x ) and L ( x ) are slowly varying right- and left-moving fields, respectively. Next, we introduce chiralbosonic fields R ( x ) ∝ e iφ ( x ) and L ( x ) ∝ e iφ ¯1 ( x ) satisfy-ing standard commutation relations [ φ r ( x ) , φ r ( x )] = irπδ rr sgn( x − x ). This ensures the correct anticom-mutation relation between fermionic operators of thesame species, while the remaining commutation relationscan be ensured by Klein factors, which we do not in-clude explicitly. It is also useful to define local con-jugate fields φ = ( φ ¯1 − φ ) / θ = ( φ ¯1 + φ ) / φ ( x ) , θ ( x )] = iπ sgn( x − x ). Small-momentuminteractions are now included via the standard kineticterm H = v π R dx { K [ ∂ x θ ( x )] + K [ ∂ x φ ( x )] } , where v is the velocity and K the Luttinger liquid parameter. Furthermore, momentum-conserving multi-electron pro-cesses involving backscatterings can lead to the openingof gaps when relevant in the renormalization group (RG)sense.
In our case, to lowest order in the interaction,the corresponding term reads H mCDW = ˜ V m Z dx [( R † L ) m e iα + H . c . ] , (2)where we have neglected quickly oscillating contribu-tions. Here, ˜ V m ∝ V m g m − B , where g B is the strength of the backscattering term induced by interactions. Interms of the bosonic fields, the CDW term takes the form H mCDW = R dx H mCDW ( x ) with H mCDW ( x ) = − | ˜ V m | (2 πa ) m cos(2 mφ ( x ) + α − α ) , (3)where a is a short-distance cutoff and α an irrelevantphase shift. The above term is of sine-Gordon form andopens a full gap at the Fermi level whenever relevant inthe RG sense. This can be achieved if K < /m or if thebare coupling constant is already of order one comparedto the Fermi energy. From now on, we therefore focus onthe case where H mCDW is relevant.We now calculate the FBC in a semi-infinite systemwith a single boundary at x = 0. In the semiclassicallimit of infinitely strong pinning, the bosonic field φ takesa constant value in order to minimize the cosine term.Explicitly, we find − φ ( ∞ ) = ( α − α ) / (2 m ) + pπ/m ,where p is an integer. At the edge of the system at x = 0,on the other hand, we impose vanishing boundary con-ditions by demanding R (0) + L (0) = 0. This implies φ (0) − φ ¯1 (0) = − φ (0) = π mod 2 π . Using that theelectron density is given by ρ ( x ) = − ∂ x φ ( x ) /π , we canthen calculate the charge located at the boundary of thesystem as Q B = − [ φ ( ∞ ) − φ (0)] /π in units of the elec-tron charge e . Plugging in the bulk and edge values for φ found above, we obtain for Q DB ≡ Q B (up to an irrel-evant constant) Q DB = α πm + pm mod 1 . (4)This result has several interesting features: Firstly, wesee that the FBC is a linear function of α with a slope1 / πm . For m = 1, this agrees with the result thatwas previously obtained for noninteracting systems, but the derivation presented here also holds in thepresence of interactions. Secondly, for fixed α , thereare m different values for the FBC, Q DB − α/ πm ∈{ , /m, ..., ( m − /m } . For m >
1, we therefore findthat the ground state is m -fold degenerate. Thirdly, thesedifferent ground states cannot be connected to one an-other under adiabatic evolution of α . As such, a givenbranch of the FBC is 2 πm -periodic, while the Hamil-tonian is 2 π -periodic. Finally, we emphasize that theseresults are independent of the exact value of K but holdwhenever H mCDW is relevant.In fact, Eq. (4) can also be understood from moregeneral arguments without the use of the bosonizationformalism. To see this, let us assume that the bulk isfully gapped by the backscattering mechanism discussedabove. If we shift the origin of the system by π/mk F ≡ λ F / m , the FBC cannot change. Furthermore, any shiftof the lattice by d can always be compensated by shifting α . Thus, the FBC is a function of both of them and nec-essarily has the form Q B ( α, d ) ≡ Q B ( α/ π + 2 md/λ F ).This is nothing but a form of ‘Galilean invariance’ in α and d . On the other hand, a shift by d changes Q B by FIG. 2. Pictorial representation of the ground state for m = Z = 3 and a certain phase α in the investigated regime.Every 9th site is occupied by a particle, which is indicated bythe small arrows pointing up. The ground state is three-folddegenerate as one can move the particles to the green or blueunit cells. d ¯ ρ B , where ¯ ρ B = 2 /λ F is the average bulk density. Thus,we find that the FBC is a linear function of not only d butalso α and has the form Q B = ( α/ π + 2 md/λ F ) /m + C ,where C is a constant. Again, we find that the slope ofthe phase dependence is 1 / πm . Simultaneously, theremust be m different branches of the FBC [correspond-ing to m values C = 0 , /m, ..., ( m − /m ] since H is2 π -periodic. Effective model.
To illustrate the m -fold degenerateground state and the phase dependence of the FBC [seeEq. (4)], we consider the following tight-binding modelwith N s sites H = − t N s − X n =1 ( a † n a n +1 + H . c . ) + r X l =1 N s − l X n =1 U l ˆ ρ n ˆ ρ n + l + N s X n =1 v ex cos (cid:18) πZ n + α (cid:19) ˆ ρ n , (5)where ˆ ρ n = a † n a n , t is the hopping amplitude, v ex the am-plitude of the potential modulation with period Z andphase α , and U l the electron-electron interaction withrange r . We choose U l = U and r = Z ( m − ρ B = mZ is the average bulk density (corresponding tofilling ν = m ). For sufficiently large v ex or U , this meansthat every m th minimum is occupied in the thermody-namic limit (see Fig. 2). The ground state is then m -folddegenerate because one could shift all particles simulta-neously to the next minimum.Next, we investigate the evolution of the FBC with α for the three ground states in the Z = m = 3case. We calculate Q B with perturbation theory for U (cid:29) v ex (cid:29) t , see Supplemental Material (SM) fordetails, and compare these results with results obtainedfrom a numerically exact density matrix renormalizationgroup (DMRG) approach. For an open system of finitesize one gets a larger degeneracy of the ground state asone can also shift single particles close to the boundaries.Thus, we do not use an integer number of unit cells butcut some sites at the boundary. The ground state of thesystem is then nondegenerate and we can perform a vari-ational ground state search. For more details we refer tothe SM. We summarize our results in Fig. 3.We confirm, in accordance with Eq. (4), that Q B shows FIG. 3. FBC Q B of the effective model with m = Z = 3 and t = 1. Main panel: Q B as a function of α at v ex = 5 and U =10 calculated with DMRG (crosses) and perturbation theory(solid lines). For details about the calculation see SM. TheFBC shows a linear slope 1 / π up to a periodic function anda jump of size unity. Inset: False color plot showing δQ B = Q B (2 π ) − Q B (cid:0) π (cid:1) for different v ex and U . The expectedlinear slope (white region) is observed already for relativelysmall v ex and intermediate U . a linear slope 1 / πm = 1 / π up to a 2 π/Z -periodic func-tion (with Z = 3). The inset of Fig. 3 shows a false colorplot demonstrating for which values of v ex and U thislinear slope indicated by δQ B = Q B (2 π ) − Q B (cid:0) π (cid:1) = (white region) is stabilized. We observe that this is al-ready the case for relatively small v ex and intermediate U . The general phenomenology of fractionally quantizedslopes in the FBC of this 1D model is therefore quitegeneral and does not require fine tuning. The additionalmodulation by a 2 π/Z -periodic function that was notpresent in Eq. (4) is a consequence of commensurabilitybetween the lattice constant and the Fermi wavelengthand vanishes in the continuum limit. FBC in two dimensions - FQHE.
Next, we study theFBC in a 2DEG in the FQHE regime at odd filling fac-tors ν = 1 / (2 l + 1) for an integer l . To facilitate theanalytical treatment of strong interactions, we make useof a coupled-wire construction of the FQHE. Weconsider an array of N parallel nanowires, where theindividual wires are oriented along the x axis and thewires are stacked along the y axis. We assume peri-odic boundary conditions along the latter, realizing thecylinder geometry shown in Fig. 4. The kinetic term is H = P n H ,n with H ,n = − ~ m ∗ R dx Ψ † n ( x ) ∂ x Ψ n ( x ),where Ψ † n ( x ) [Ψ n ( x )] creates [annihilates] a spinless elec-tron of mass m ∗ at the position x in the n th wire. Amagnetic field B is applied perpendicular to the surfaceof the cylinder, with the corresponding vector potentialchosen along the y axis, A = Bx ˆ y . Additionally, anexternal flux Φ is created by a second magnetic field B oriented along the x axis. The corresponding vec-tor potential is chosen along the y axis, A = ( B R/ y ,where R = N a y / π is the radius of the cylinder and a y denotes the interwire distance. The total flux throughthe cylinder is then given by Φ = πR B . Finally, thetunneling between neighboring nanowires is described by H T = P n H T,n +1 / with H T,n +1 / = te iϕ Z dx e ik B x Ψ † n +1 Ψ n + H . c . (6)Here, k B = eBa y / ~ and ϕ = e ~ Φ N . To treat interactions,we again linearize the spectrum around the Fermi points,Ψ n ( x ) = R n ( x ) e ik F x + L n ( x ) e − ik F x , and switch to abosonized language by writing R n ( x ) ∝ e iφ n ( x ) , L n ( x ) ∝ e iφ ¯1 n ( x ) with [ φ rn ( x ) , φ r n ( x )] = irπδ rr δ nn sgn( x − x ).Small-momentum interactions can then be included inthe standard way and lead to a gapless sliding Luttingerliquid phase that we will not characterize here. Forour purposes, it suffices to note that if k F becomes com-mensurable with k B such that 2 k F /k B = ν , an additionalmomentum-conserving multi-electron process can be con-structed such that a gap is opened at the Fermi level andthe FQHE at filling ν is realized. Explicitly, the termthat opens the gap is H lT,n +1 / = R dx H lT,n +1 / with H lT,n +1 / = t l e iϕ R † n +1 L n ( R † n +1 L n +1 ) l ( R † n L n ) l + H . c . (7)Here, t l ∝ tg lB , where g B is the strength of the backscat-tering term induced by interactions. Introducing thefields η rn = ( l + 1) φ rn − lφ ¯ rn , Eq. (7) becomes H lT,n +1 / = − | t l | (2 πa ) l +1 cos( η n +1) − η ¯1 n − ϕ + ϕ ) , (8)where ϕ is an irrelevant phase shift. In this represen-tation, it is evident that all fields are pinned pairwise,such that the system is indeed fully gapped given that H lT,n +1 / is the leading relevant term. This can alwaysbe achieved for a suitable set of interaction parameters. The case l = 0 corresponds to the IQHE with ν = 1,where the gap is opened even without interactions.We now calculate the FBC in dependence on the fluxΦ. At low energies, the argument of the cosine termin Eq. (8) is pinned to a constant value. Taking thesum over all wires n = 1 , ..., N , this pinning implies − P n φ n ( ∞ ) = N ν ( ϕ − ϕ ) / pνπ in terms of thelocal fields φ n = ( φ ¯1 n − φ n ) /
2. Here, p is an integer.On the other hand, imposing vanishing boundary con-ditions at x = 0 leads to P n φ n (0) = − N π/ π .Writing the 2D FBC as Q DB = P n Q DB,n , where Q DB,n = − [ φ n ( ∞ ) − φ n (0)] /π is the FBC in the n th wire, the abovegives us (up to an irrelevant constant) Q DB = Φ ν π e ~ + pν mod 1 . (9) FIG. 4. An array of nanowires tunnel-coupled by t arrangedto a cylinder threaded by an external flux Φ. A magnetic field B perpendicular to the cylinder surface drives the system intoan FQHE phase. The FBC Q DB is calculated in the red regionand shown to vary linearly with Φ, see Eq. (9). Thus, the FBC has a linear slope in Φ which is quantizedin units of νe/h . At fractional filling ν = 1 / (2 l + 1) with l >
0, this slope is (2 l + 1) times smaller than in theIQHE case l = 0. Furthermore, there are 2 l + 1 differentbranches of the FBC that cannot be connected underadiabatic evolution of Φ. Finally, the Hall conductancecan be obtained from the FBC following Ref. 33, yielding σ xy = e ˙ Q DB / ˙Φ = e ν/h as expected. Experimental signatures.
The sample geometry de-scribed above can be realized by a Corbino disk, seeFig. 1. The FBC is then accessible in a rather straight-forward way using, e.g., STM techniques to measurethe charge located at the boundary of the disk while theexternal flux Φ is varied. This allows for several interest-ing ways to probe the FQHE: Firstly, observing the slopeof the linear flux dependence allows one to probe the fill-ing factor, see Eq. (9). This can further be corroboratedby observing the evolution of the FBC as Φ is varied adi-abatically. The FBC will then be (2 π/ν )-periodic in Φ,with a jump of size unity occurring at a particular valueof Φ. Secondly, the different branches of the FBC canbe connected if fractional charges are allowed to tunnelbetween opposite boundaries due to, e.g., a constriction,see again Fig. 1. By measuring the FBC repeatedly inthe presence of a constriction, one finds that it can take2 l + 1 different values, reflecting the (2 l + 1)-fold groundstate degeneracy. Similarly, if one now observes the evo-lution of the FBC with Φ, also jumps of fractional size s/ (2 l + 1), where s = 1 , ..., l is another integer, can beobserved when the system switches from one ground stateto another. We note that due to translational invarianceit suffices to measure the FBC along a small part of theboundary rather than along the entire circumference, seeFig. 1. In this case, instead of measuring the absolutevalues of the slopes and jumps of the FBCs, one shouldmeasure their ratios for different filling factors, whichagain become universal. Conclusions.
We have studied FBCs in strongly inter-acting CDW-modulated nanowires and in Corbino disksin the FQHE regime at odd filling factors threaded byan external flux. In both cases, the FBC displays uni-versal features that do not depend on microscopic detailsof the models such as the exact values of the interac-tion parameters. In the nanowire (FQHE) case, we findthat the FBC depends linearly on the phase offset (flux)with a quantized slope that is determined by the fillingfactor. Furthermore, the different possible values of theFBC at a fixed phase offset (flux) label different degener-ate ground states that cannot be adiabatically connected.The observation of these features is well within experi-mental reach and opens up a promising route to probestrongly interacting phases via FBCs.As an outlook, we note that our findings can readily beextended to more general filling factors ν = k/ (2 l + 1),where k is an integer that is coprime to 2 l + 1. Theslope of the linear flux dependence will then again bedetermined by ν , and a given branch of the FBC will be2 π (2 l + 1)-periodic under adiabatic evolution of Φ with k jumps of size unity occurring at specific values of Φ. Acknowledgments.
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Composite Fermions (Cambridge UniversityPress, Cambridge, 2007). B. I. Halperin, Phys. Rev. B , 2185 (1982). D. A. Syphers, K. P. Martin, and R. J. Higgins, Appl.Phys. Lett. , 293 (1986). P. F. Fontein, J. M. Lagemaat, J. Wolter, and J. P. Andr´e,Semicond. Sci. Technol. , 915 (1988). V. T. Dolgopolov, A. A. Shashkin, N. B. Zhitenev, S. I.Dorozhkin, and K. von Klitzing, Phys. Rev. B , 12560(1992). M. J. Zhu, A. V. Kretinin, M. D. Thompson, D. A. Ban-durin, S. Hu, G. L. Yu, J. Birkbeck, A. Mishchenko, I. J.Vera-Marun, K. Watanabe, T. Taniguchi, M. Polini, J. R.Prance, K. S. Novoselov, A. K. Geim, and M. Ben Shalom,Nat. Commun. , 14552 (2017). B. A. Schmidt, K. Bennaceur, S. Gaucher, G. Gervais, L.N. Pfeiffer, and K. W. West, Phys. Rev. B , 201306(R)(2017). In an experimental realization, it will be more convenientto tune k F by varying the filling of the wire rather thanadjusting the period of the potential. The CDW gap will, however, be renormalized by the in-teraction, see Ref. 60. At this point, it is crucial that we work with odd fillingfactors. At even filling factors, the presence of multiplecompeting momentum-conserving terms can lead to a morecomplicated behavior that is not captured here. While Ref. 33 studied the IQHE, the derivation of the Hallconductance presented there remains valid also in the pres-ence of interactions. upplemental Material: “Fractional boundary charges with quantized slopes ininteracting one- and two-dimensional systems”
Katharina Laubscher, Clara S. Weber, Dante M. Kennes,
2, 3
MikhailPletyukhov, Herbert Schoeller, Daniel Loss, and Jelena Klinovaja Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland Institut f¨ur Theorie der Statistischen Physik, RWTH Aachen, 52056 Aachen,Germany and JARA - Fundamentals of Future Information Technology Max Planck Institute for the Structure and Dynamics of Matter,Center for Free Electron Laser Science, 22761 Hamburg, Germany
GROUND STATE FOR FINITE AND OPENSYSTEM
When calculating the ground state of the effective one-dimensional model [Eq. (5) of the main text] for an openand finite system with a fixed number of particles N p , onegets a huge degeneracy due to the missing particles at thesystem boundaries. To avoid this degeneracy we cut thesystem and take only N s = mZN p − ( m − Z = 9 N p − N s = 9 N p sites (for Z = 3 and m = 3).The degeneracy is then lifted and there is only one possi-ble ground state. This procedure corresponds to forcingthe last 6 sites to be empty. The other two possibleground states that would occur in the thermodynamiclimit can then be found by putting either 3 or all 6 emptysites to the other boundary of the chain. We will use thisprocedure for our DMRG calculations as well as for theanalytical calculations of the boundary charge.Using this ground state search, one gets a periodicityof 2 π . To get the periodicity of 6 π , we calculate all threeground states. We expect these states to evolve into eachother when executing an adiabatic time evolution in thegrand-canonical ensemble with the chemical potential lo-cated in the charge gap. One then gets the periodicityof 6 π which we show in the main text. For convenience,we choose the chemical potential in such a way that thejumps of the adiabatic time evolution and the ones ofthe ground state search occur at the same position. Thepositions of the jumps in the adiabatic time evolutionmay change slightly when changing the chemical poten-tial within the gap.The average of the FBC is given by Q B = ¯ N s X n =1 f n ( ρ n − N p / ¯ N s ) , (S1)where ¯ N s is the number of sites including the 6 emptysites, N p is the number of particles, and ρ n = h a † n a n i .The envelope function is denoted by f n and needs to de-cay smoothly from 1 to 0. For our numerical calculationswe take a linear slope for the decay of length l p . The cen-ter of this slope has a distance of L p to the left boundary.For the calculations shown in Fig. 3 of the main text weuse N s = 174 ( ¯ N s = 180) with L p = 90 and l p = 90. ANALYTICAL CALCULATION OF THE FBC
In this section we calculate the FBC for the effectiveone-dimensional model in dependence of the phase α an-alytically. We focus on the case of Z = 3 and m = 3(other cases can be treated analogously) and consider theatomic limit with strong electron-electron (Coulomb) in-teraction U l = U (cid:29) v ex (cid:29) t . We introduce an effectiveunit cell of Z eff = Zm = 9, so that the average bulkdensity is ¯ ρ B = Z eff = .In the atomic limit the problem of finding the FBC inthe given strongly interacting model can be reduced toan effective single-particle model, in which a particle canoccupy one of the first m sites of the effective unit cellwith Z eff sites. As shown in Ref. 1 the FBC in this limitis dominantly given by the polarization contribution deepin the bulk, which has the form Q B ≈ Q P = − Z eff Z eff X j =1 j ( ρ bulk j − ¯ ρ B ) (S2)= − Z eff m X j =1 jρ bulk j + Z eff + 12 Z eff (S3)= − m X j =1 jρ bulk j + 59 . (S4)Depending on the minima of the cosine potential (seeFig. S1), a particle can sit either on site j = 1 (for 0 <α < π ), or on j = 2 (for π < α < π ), or on j = 3 (for π < α < π ). We thus get three plateaus, Q B (cid:18) < α < π (cid:19) ≈ −
19 + 59 = 49 , (S5) Q B (cid:18) π < α < π (cid:19) ≈ −
29 + 59 = 39 , (S6) Q B (cid:18) π < α < π (cid:19) ≈ −
39 + 59 = 29 . (S7) Vicinity of α = 0 , π , π At α = 0 , π , π the minima contain two sites with thesame onsite potential. Therefore we consider the first a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n π πα − v j j = 1 j = 2 j = 3 FIG. S1. On-site potentials v j ( α ) = cos( π j + α ) for j =1 , , order degenerate perturbation theory in t in the threedifferent intervals around these values.1) π < α < π : | ψ i ≈ c (1)1 | i + c (1)3 | i . Then we find Q B (cid:16) π < α < π (cid:17) ≈ − (cid:16) | c (1)1 | + 3 | c (1)3 | (cid:17) + 59 (S8)= − (cid:16) −| c (1)1 | + | c (1)3 | (cid:17) + 39 . (S9)However, this formula is incorrect, because the hybridiza-tion between | i and | i is in O ( t ) impossible, and weneed to revise the above result.Consider the two subintervals 1a) π < α < π ; 1b) π < α < π .1a) For π < α < π the density is mostly located on | i , with a small admixture of | i , which replaces | i inEq. (S9). Thus the correct expression reads Q B (cid:18) π < α < π (cid:19) ≈ − (cid:16) | c (1)1 | + 0 | c (1)3 | (cid:17) + 59 ≈ − (cid:16) | c (1)1 | − | c (1)3 | (cid:17) + 12 . (S10)1b) For π < α < π the density is mostly located on | i , with a small admixture of | i , which replaces | i inEq. (S9). Thus the correct expression reads Q B (cid:18) π < α < π (cid:19) ≈ − (cid:16) | c (1)1 | + 3 | c (1)3 | (cid:17) + 59 ≈ − (cid:16) | c (1)1 | − | c (1)3 | (cid:17) + 16 . (S11)Comparing Eqs. (S10) and (S11) and taking into accountthat | c (1)1 | − | c (1)3 | is a continuous function of α (seebelow) vanishing at α = π , we observe that at this valueof α the boundary charge value jumps from to , suchthat the jump value is − = − . 2) π < α < π : | ψ i ≈ c (2)3 | i + c (2)2 | i . This gives us Q B (cid:18) π < α < π (cid:19) ≈ − (cid:16) | c (2)3 | + 2 | c (2)2 | (cid:17) + 59 ≈ − (cid:16) | c (2)3 | − | c (2)2 | (cid:17) + 518 . (S12)3) − π < α < π : | ψ i ≈ c (3)2 | i + c (3)1 | i , leading to Q B (cid:16) − π < α < π (cid:17) ≈ − (cid:16) | c (3)2 | + | c (3)1 | (cid:17) + 59 ≈ − (cid:16) | c (3)2 | − | c (3)1 | (cid:17) + 718 . (S13)The coefficients c a and c b are found from the eigenvalueproblem (cid:18) v a − v b − t − t − v a − v b (cid:19) (cid:18) c a c b (cid:19) = − s(cid:18) v a − v b (cid:19) + t (cid:18) c a c b (cid:19) (S14)and it follows c b = v a − v b s(cid:18) v a − v b (cid:19) + t c a t , (S15) c a = t (cid:20) v a − v b + q(cid:0) v a − v b (cid:1) + t (cid:21) + t , (S16) c a − c b = t − (cid:20) v a − v b + q(cid:0) v a − v b (cid:1) + t (cid:21) (cid:20) v a − v b + q(cid:0) v a − v b (cid:1) + t (cid:21) + t (S17)= − ( v a − v b ) v a − v b + q(cid:0) v a − v b (cid:1) + t (cid:20) v a − v b + q(cid:0) v a − v b (cid:1) + t (cid:21) + t . (S18)The results are shown in Fig. S3 for certain parameters,where we also compare them to DMRG data.When we made the ansatz that we only need one par-ticle in a cell of Z eff = 9 sites, we assumed that thereare always two empty minima between the particles dueto the repulsive electron-electron interaction. However,it is possible to have configurations where there is onlyone empty minimum between two particles. Then, bothparticles need to be located on the outer site of theirminimum as shown in Fig. S2. In this case they also donot ‘see’ each other’s Coulomb interaction and it would ↑ ↑· · · · · · | {z } Range of Coulomb interaction of the first particle | {z }
Range of Coulomb interaction of the second particle
FIG. S2. States that are ground states at t = 0 but can beneglected at t = 0. Two of the particles have only a distanceof ( m −
1) minima. The energy is higher than for the stateswhere every m th minimum is occupied as the coupling of thegreen particle to the right and of the blue one to the left ismuch smaller in these cases. be a ground state for t = 0. However, we do not needto consider them for the case with t = 0. Indeed, theneglected states are not coupled to the used ones in theorders that we look at. So there are no neglected cou-plings. Additionally, all states that have a contributionof those new states should have a larger energy than thecalculated ground state because the coupling to the adja-cent site is much smaller for the configurations shown inFig. S2 due to the Coulomb interaction. Therefore, thesestates cannot contribute to the ground state for t = 0.Using this degenerate perturbation theory, we findsome discontinuities at α = π , π, π (see Fig. S3) becausein the vicinity of these points, there are not two sites inthe minimum. In the next section we will remove thesediscontinuities by treating the vicinities of these pointsin second order in t with a non-degenerate perturbationtheory. Vicinity of α = π , π, π In the vicinity of these points we can use non-degenerate perturbation theory where, up to first orderin the perturbation, the ground state is given by | Ψ i = | n i + X m = n V mn E n − E m | m i . (S19)Here, | n i denotes the ground state for t = 0, and V mn are the matrix elements between the ground state andthe excited states m , which are given by the hopping inour model.In the given regions there is one site in each minimumof the on-site potential. We will call this site b , while thetwo adjacent sites will be called a and c . We then get | Ψ i = | b i + tv c − v b | c i + tv a − v b | a i (S20)for the ground state. Taking into account that this state is not normalized, we get | c b | = 1 − (cid:18) tv a − v b (cid:19) − (cid:18) tv c − v b (cid:19) + O (cid:0) t (cid:1) , (S21) | c a | = (cid:18) tv a − v b (cid:19) + O (cid:0) t (cid:1) , (S22) | c c | = (cid:18) tv c − v b (cid:19) + O (cid:0) t (cid:1) . (S23)The boundary charge in the three different regions canthen be calculated as follows: Q B (cid:18) < α < π (cid:19) ≈ − (cid:16) | c (1)3 | + | c (1)1 | + 2 | c (1)2 | (cid:17) + 59 ≈ (cid:16) −| c (1)3 | + | c (1)2 | (cid:17) + 49 , (S24) Q B (cid:18) π < α < π (cid:19) ≈ − (cid:16) | c (2)2 | + 3 | c (2)3 | + 4 | c (2)1 | (cid:17) + 59 ≈ (cid:16) −| c (2)2 | + | c (2)1 | (cid:17) + 29 , (S25) Q B (cid:18) π < α < π (cid:19) ≈ − (cid:16) | c (3)1 | + 2 | c (3)2 | + 3 | c (3)3 | (cid:17) + 59 ≈ (cid:16) −| c (3)1 | + | c (3)3 | (cid:17) + 39 . (S26)We then insert | c c | − | c a | ≈ (cid:18) tv c − v b (cid:19) − (cid:18) tv a − v b (cid:19) (S27)to get the final result that is shown in Fig. S3 togetherwith the results of the first order perturbation theorycalculated above. Uniting the results
In the previous sections we calculated the behavior ofthe boundary charge in different regimes of α . To get afinal expected curve, one needs to decide where to changebetween those regimes. Basically we have two differentfunctions for the boundary charge. One should be validaround α = π/ , π, π/ α =0 , π/ , π/
3. These two results are plotted in the wholeinterval of [0 , π ] in Fig. S3.As one can see, both results fit a certain part of the nu-merical curve quite well while there are other parts where π π π α . . . . Q B DMRGperturbation theory 1perturbation theory 2
FIG. S3. Boundary charge as a function of α for the effectivemodel calculated with DMRG and perturbation theory. Theother parameters are m = 3, Z = 3, t = 1, v ex = 5 and U l = U = 10 for l = 1 , ...,
6. We used N s = 174 and L p = 90, l p =90 for the envelope function to get the DMRG results. Thedashdotted line shows the results calculated in the vicinityof α = π , π, π , while the dashed line was calculated in thevicinity of α = 0 , π , π . they show useless behavior like jumps and divergences.Nevertheless they coincide very well in the intermediateregions between the regimes where they were calculated.To get one final analytical curve the method of calcula-tion was changed at the points where both curves crosseach other. The final result can be seen in Fig. S4. Thenumerical and analytical results lie nearly perfectly ontop of each other. Fig. S4 corresponds to a zoom intoFig. 3 of the main text, where we show all three groundstates with their periodicity of 6 π . There, we find a jumpof unity for each of the ground states because a particleleaves the system at that point. In Fig. S4 we see onlya jump by 1/3 because the system changes to anotherground state as indicated by the colors. Thereby, allparticles are shifted by one minimum to get into the newreal ground state of our system. As already mentionedabove, the other two ground states can be found by forc-ing other sites to have zero occupation. For the analyticalcalculation this means that sites 4 , , , , , ,
3. The boundary charge is thenchanged by − / − / Limits of our perturbation theory
As we performed the perturbation theory in the regime U (cid:29) v ex (cid:29) t , we expect it to fail when U and v ex arenot large enough. In Fig. S5 we calculate the boundarycharge in dependence of α for different U and v ex withconstant U/v ex = 2. For large values of U and v ex theresults coincide very well with our numerical DMRG re-sults. For smaller values of U and v ex the curves startto differ. When U and v ex are of the order of t the per-turbation theory does not even give us a smooth curve. π π π α . . . . Q B DMRGperturbation theory
FIG. S4. Boundary charge as a function of α for the effectivemodel calculated with DMRG and perturbation theory. Theother parameters are m = 3, Z = 3, t = 1, v ex = 5 and U l = U = 10 for l = 1 , ...,
6. We used N s = 174 and L p = 90, l p = 90 for the envelope function to get the DMRG results.The two curves of Fig. S3 are now combined to one final curve. α b) α . . Q B a) DMRGpert. theory α d) α . . Q B c) π π π α f) π π π α . . Q B e) FIG. S5. Results of DMRG and perturbation theory for t =1 . v ex , U ) = a) (5 . , . . , . . , . . , . . , . . , . v ex and U as expected,while there are clear drawbacks for smaller v ex and U . TheDMRG results are obtained with N s = 174, L p = 90, and l p = 90. For those parameters the results of the different regimesof α do not agree in the intermediate region and cannotbe united in a satisfying way. DEPENDENCE ON ENVELOPE FUNCTION
In the thermodynamic limit the boundary charge needsto be independent of the details of the envelope function.However, the boundary charge can slightly depend onthe envelope function for finite system sizes as shown in .
001 0 .
002 0 .
003 0 .
004 0 . / ¯ N s . . . Q B ( π ) − Q B (cid:0) π (cid:1) L p = ¯ N s / , l p = ¯ N s / L p = ¯ N s / , l p = ¯ N s L p = ¯ N s / , l p = ¯ N s / / FIG. S6. Q B (2 π ) − Q B (cid:0) π (cid:1) for different system sizes anddifferent envelope functions. ¯ N s = N s + ( m − Z = N s + 6describes the system size including the blocked sites wherethe density is forced to be zero. For ¯ N s → ∞ all curves tendto 1 /
9. The other parameters are t = 1, v ex = 3, and U = 3. Fig. S6. To be as close as possible to the thermodynamiclimit, we choose the envelope function with L p = ¯ N s / l p = ¯ N s / N s = N s + 6 denotes the system includingthe sites that we forced to be empty. With this choicealready systems of relatively small size give us a valueof Q B (2 π ) − Q B (cid:0) π (cid:1) that coincides with the one in thethermodynamic limit. INTERFACE BETWEEN TWO CDWS
In this section, we show how the analytical argumentspresented in the main text can be extended to describethe charge located at the interface between two CDWsin a 1D nanowire.We consider an interface between two CDWs describedby a spatially modulated potential of the form V m ( x ) = ( V m cos(2 mk F x + α < ) , x < , V m cos(2 mk F x + α > ) , x > , (S28)where α < ( α > ) describes the phase offset of the CDWin the domain x < x > x ∈ ( −∞ , x ∈ (0 , + ∞ ) separately. In terms of the conjugate bosonic fields φ and θ , the CDW term then takes theform H mCDW = R dx H mCDW ( x ) with H mCDW ( x ) = ( − | ˜ V m | (2 πa ) m cos(2 mφ ( x ) + α < − α ) , x < , − | ˜ V m | (2 πa ) m cos(2 mφ ( x ) + α > − α ) , x > , (S29)where α is again an irrelevant overall phase shift. TheCDW term is minimized for the pinning values φ ( x ) = ( − ( α < − α ) / m + lπ/m, x < , − ( α > − α ) / m + nπ/m, x > , (S30)where l and n are integers. Therefore, the charge locatedat the interface is given by Q D = − Z + ∞−∞ dx ∂ x φ ( x ) π = − π [ φ (+ ∞ ) − φ ( −∞ )]= ( l − n ) /m + ( α > − α < ) / πm mod 1 . (S31)We thus find that the fractional charge changes linearlywith the phase difference α > − α < with a slope of 1 / πm .Finally, we note that analogous considerations allowus to recover the fractional charge of the excitations inthe 2D case. Indeed, a bulk excitation in the 2D FQHEcorresponds to a kink (domain wall) in the pinned combi-nation of the fields η n +1) − η ¯1 n for a given n , see Eq. (8)in the main text, while the uniform phase ϕ drops out.By using P n ( η n +1) − η ¯1 n ) = − l + 1) P n φ n andusing that the charge density of a single wire is givenby ρ n = − ∂ x φ n ( x ) /π in units of the electron charge e ,we find that a kink between two adjacent minima of thecosine carries the charge e/ (2 l + 1). [1] M. Pletyukhov, D. M. Kennes, K. Piasotski, J. Klinovaja,D. Loss, and H. Schoeller, Phys. Rev. Research2