Topological transport of mobile impurities
D. Pimenov, A. Camacho-Guardian, N. Goldman, P. Massignan, G. M. Bruun, M. Goldstein
TTopological transport of mobile impurities
D. Pimenov, ∗ A. Camacho-Guardian,
2, 3
N. Goldman, P. Massignan, G. M. Bruun, and M. Goldstein William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA Department of Physics and Astronomy, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark T.C.M. Group, Cavendish Laboratory, University of Cambridge,JJ Thomson Avenue, Cambridge, CB3 0HE, U.K. Universit´e Libre de Bruxelles, CP 231, Campus Plaine, 1050 Brussels, Belgium Department de F´ısica, Universitat Polit`ecnica de Catalunya,Campus Nord, B4-B5, E-08034 Barcelona, Spain Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv 6997801, Israel
We study the Hall response of topologically-trivial mobile impurities (Fermi polarons) interactingweakly with majority fermions forming a Chern-insulator background. This setting involves a richinterplay between the genuine many-body character of the polaron problem and the topologicalnature of the surrounding cloud. When the majority fermions are accelerated by an external field,a transverse impurity current can be induced. To quantify this polaronic Hall effect, we computethe drag transconductivity, employing controlled diagrammatic perturbation theory in the impurity-fermion interaction. We show that the impurity Hall drag is not simply proportional to the Chernnumber characterizing the topological transport of the insulator on its own – it also depends con-tinuously on particle-hole breaking terms, to which the Chern number is insensitive. However, whenthe insulator is tuned across a topological phase transition, a sharp jump of the impurity Hall dragresults, for which we derive an analytical expression. We describe how the Hall drag and jump canbe extracted from a circular dichroic measurement of impurity excitation rates, particularly suitedfor ultracold gas experiments.
I. Introduction
The concept of topology, which asserts that certain ob-servables only depend on global geometrical characterist-ics rather than on microscopical properties, is a powerfultool in modern physics [1]. At its core, this concept isa non-interacting one: the canonical topological invari-ants, which are related to transport coefficients such asthe transverse (Hall) conductivity via the famous TKNNformula [2], are formulated in terms of single-particlewave-functions. To marry topology with interaction phe-nomena is a challenging ongoing effort [3, 4].A controllable route to the inclusion of interactions isto consider a non-interacting “majority” system, coupleit to a small number of quantum impurities, and studyinteraction effects on the impurities only. If the major-ity system is a conventional metal, the impurities aretransformed into so-called Fermi polarons [5], which bynow are routinely observed in ultracold-gas [6–10] andcondensed matter experiments [11] – for a review of theliterature, see for instance Refs. [12–14].In the context of topology, it is natural to ask ifand how topologically trivial impurities inherit the to-pological properties of the majority. This question haspreviously been addressed mainly from two perspect-ives: Either interaction effects are strong such that animpurity-majority bound state is formed [15–18], whichessentially carries the topological quantum numbers ofthe majority, or, alternatively, one can study the prob-lem in weak coupling [19], as previously done by some of ∗ [email protected] us, with the majority forming a Chern insulator. Thisperturbative approach is well-controlled and does not re-quire additional regularization.In this work we significantly expand on the weak coup-ling approach of Ref. [19]. With mostly analytical meanswe investigate the impurity Hall drag for a generic (con-tinuum) Dirac model of a Chern insulator, inspired byRef. [20]. The analytical treatment is further validatednumerically for the Haldane lattice model [21]. In con-trast to the strong-coupling limit, we find that there isno simple relation between majority transport and im-purity Hall drag, as the latter depends on an additionalparameter quantifiying the particle-hole asymmetry ofthe majority bands. Remarkably, the Hall drag simplyvanishes in the particle-hole symmetric case, reminiscentof Coulomb drag in two-layer systems [22–24]. If particle-hole symmetry is broken, the impurity Hall drag can benon-vanishing even if the majority Chern insulator is inthe trivial phase. However, upon tuning the insulatoracross its topological phase transition, the impurity Halldrag exhibits a sharp jump related to the change of themajority Chern number. This is the only clear manifesta-tion of topology in weak-coupling impurity transport. Weshow how to detect the Hall drag and jump in ultracoldgas experiments via circular dichroism, that is, measur-ing impurity excitation rates upon driving the systemwith left and right circularly polarized fields [25–28]. Asystematic method of computing the excitation rates inan interacting many-body system is presented along theway.The remainder of this paper is structured as follows:In Sec. II we present the continuum Dirac model and theevaluation of the impurity drag. In Sec. III, we invest- a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b igate the jump across the topological phase transition.The drag including its jump at the topological transitionis analyzed for the Haldane model in Sec. IV. The dichroicmeasurement is detailed in Sec. V, and conclusions andoutlook are presented in Sec. VI. Some technical detailsare relegated to Appendices. II. Drag transconductivity in the continuum model
We start by computing the impurity drag in a gen-eric continuum model and consider the following two-dimensional Bloch Hamiltonian for majority particles in-dexed by a pseudospin ↑ : H ↑ ( k ) = (cid:88) i =0 ψ †↑ ( k ) h i ( k ) σ i ψ ↑ ( k ) , (1) ψ ↑ ( k ) = ( c ↑ ,A ( k ) , c ↑ ,B ( k )) T ,h ( k ) = k x , h ( k ) = k y , h ( k ) = m + d k ,h ( k ) = d k , k = | k | , with σ = and σ i with i = 1 , , (cid:126) = c = e = 1; all quantities are measuredin appropriate powers of the (inverse) physical fermionmass, while momenta are rescaled by the band velocity.Equation (1) can be seen as a low-energy approxima-tion to a microscopic tight-binding Hamiltonian with atwo-sublattice structure ( A, B ) and broken time-reversalinvariance. The eigenenergies corresponding to (1) read (cid:15) ↑ ;1 , ( k ) = h ( k ) ∓ h ( k ) , h ( k ) = (cid:112) k + h ( k ) . (2)Without the terms d , d (which have physical dimen-sions (mass) − ), Eq. (1) describes a gapped Dirac conewith mass gap m . The term d serves as a UV regular-izer and makes the dispersion quadratic at higher ener-gies while preserving particle-hole symmetry, (cid:15) ↑ , ( k ) = − (cid:15) ↑ , ( k ). The symmetry is broken for finite d . Ex-emplary spectra are shown in Fig. 1(a). We assume | d | > | d | , thus the lower (upper) band is filled (empty).For general d , the Hamiltonian (1) is in the Altland-Zirnbauer class A [29], and gives rise to a quantized Chernnumber C . As shown below, it reads C = 12 π (cid:90) d k
12 ( m − d k )( k + ( m + d k ) ) / (3)= 12 [sign( m ) − sign( d )] . The integrand of Eq. (3) is nothing but the Berrycurvature F xy ( k ). As visualized in Fig. 1(b), for m → F xy ( k ) consists of a sharp half-quantized peak for k (cid:46) m ,arising from the Dirac fermions, on top of a broad back-ground from high-energy “spectator” fermions [20].As explicit in Eq. (3), C does not depend on theparticle-hole symmetry breaking parameter d . This is in (b) F xy ( k ) k C = 1 C = 0 d = 0 d = 0 . (cid:15) ↑ (0 , k x ) k x k (a) Figure 1. (a) Continuum bands for m = 0 . , d = 1 and twovalues of d (b) Berry curvature for d = − m = ± . m = ± . line with the geometrical interpretation of C as a windingnumber [30], which is independent of the term h com-muting with the Hamiltonian [31].For clarity of the later results, let us now compute C explicitly as C = − πσ xy [2, 32], with σ xy the transcon-ductivity; the conductivity quantum is σ = e / (cid:126) = 1 / π with the chosen units. In linear response, σ xy is propor-tional to the retarded current-current correlation func-tion, which may be obtained by analytical continuationfrom imaginary time: σ xy = lim ω → − iωA (cid:34) − (cid:104) ˆ J x ↑ ˆ J y ↑ (cid:105) ( i Ω) (cid:12)(cid:12)(cid:12)(cid:12) i Ω → ω + i + (cid:35) , (4)with A the system area, and ˆ J ↑ the current operators atvanishing external momentum. It is convenient to workin the diagonal band basis, introducing a diagonalizingunitary matrix U ( k ) (see App. A for details) U †↑ ( k ) H ↑ ( k ) U ↑ ( k ) = diag( (cid:15) ↑ , ( k ) , (cid:15) ↑ , ( k )) . (5)In this basis the current operator has matrix elements J x/y ↑ ( k ) = U †↑ ( k ) ∂H ↑ ( k ) ∂k x/y U ↑ ( k ) . (6)The imaginary time correlator in Eq. (4) can now be ω k , k
2; Ω + ω k , k ParticleHole J x ↑ , ( k ) J y ↑ , ( k )Ω Ω Figure 2. Diagram representing Eq. (7), with α = 1 , β = 2. written as − (cid:104) ˆ J x ↑ ˆ J y ↑ (cid:105) ( i Ω) = (7) A (cid:90) k G ↑ ,α ( ω k , k ) G ↑ ,β (Ω + ω k , k ) J x ↑ ,αβ ( k ) J y ↑ ,βα ( k ) , (cid:90) k ≡ (cid:90) d k dω k (2 π ) , G ↑ ,α ( ω k , k ) = 1 iω k − (cid:15) ↑ ,α ( k ) , where α, β refer to band indices and the Einstein summa-tion convention is implied. The standard diagrammaticalrepresentation of Eq. (7) is shown in Fig. 2. The Mat-subara Green function G ↑ , describes the propagation ofa hole in the filled lower band, while G ↑ , represents aparticle in the upper band. The frequency integral in Eq.(7) only receives contributions when α (cid:54) = β , and thus onecan view creation of virtual particle-hole pairs as the ori-gin of the conductivity. These quasiparticles are virtual,since the external field does not provide enough energy(Ω →
0) to overcome the band gap.Evaluation of Eqs. (7) and (4) is straightforward. Notethat the O (Ω ) part of (7) vanishes upon k -integration.One finds σ xy = − i (cid:90) d k (2 π ) J x ↑ , ( k ) J y ↑ , ( k ) − J x ↑ , ( k ) J y ↑ , ( k )( (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k )) = 12 π (cid:90) d k π Im (cid:104) J x ↑ , ( k ) J y ↑ , ( k ) (cid:105) ( (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k )) = − π C . (8)Inserting current matrix elements and dispersions intoEq. (8) produces Eq. (3).After this noninteracting prelude, we are ready toattack the polaron problem. We consider a minorityparticle species indexed by ↓ , with a trivial quadraticHamiltonian H ↓ ( p ): H ↓ ( p ) = (cid:15) ↓ ( p ) c †↓ ( p ) c ↓ ( p ) , (cid:15) ↓ ( p ) = p M . (9)We can view the impurities as governed by a similar tight-binding Hamiltonian as the majority, but with a chemicalpotential almost at the bottom of the lower band, aroundwhich the dispersion is approximated by an effective mass M . Higher impurity bands can be safely neglected.The majority and minority particles interact via anonsite-interaction H int [19], which does not distinguish between the sublattices (recall that the sublattices giverise to the two-band structure): H int = gA (cid:88) (cid:96) = A,B (cid:88) k , p , q c †↑ ,(cid:96) ( k + q ) c ↑ ,(cid:96) ( k ) c †↓ ( p − q ) c ↓ ( p ) = gA (cid:88) k , p , q c †↑ ,α ( k + q ) c ↑ ,β ( k ) c †↓ ( p − q ) c ↓ ( p ) W αβ ( k , q ) ,W αβ ( k , q ) ≡ (cid:104) U †↑ ( k + q ) U ↑ ( k ) (cid:105) αβ , (10)where we have rotated to the band space in the secondline. Now we imagine a constant and uniform force E = E e y acting on both majority and minority particles[33]. Due to the interaction H int , a transverse impuritycurrent J x ↓ will be induced; without interaction, there isnone due to time reversal symmetry of the impurities.To quantify this effect, we must compute the Hall dragtransconductivity σ ↓↑ ≡ lim ω → − iωA (cid:34) − (cid:104) ˆ J x ↓ ˆ J y ↑ (cid:105) ( i Ω) (cid:12)(cid:12)(cid:12)(cid:12) i Ω → ω + i + (cid:35) . (11)This computation will be done to second order in theimpurity-majority coupling g , since the first order con-tribution vanishes [19]; thus, attractive and repulsive in-teractions lead to the same result. We point out that suchperturbative expansion is well-controlled for small g , andno resummation is needed, in contrast with the recentevaluation of longitudinal polaron drag in the metalliccase [34].As in the case of Coulomb drag in two-layer systems[22], the O ( g ) contribution corresponds to the two dia-grams shown in Fig. 3. We evaluate these diagrams toleading order in the small impurity density n ↓ . The dia-grams involve an impurity loop and are therefore propor-tional to n ↓ , unlike the single-particle polaron diagramswhich have an impurity “backbone” [35]. It is convenientto identify the impurity lines that represent filled states( ˆ= impurity holes). Since these carry vanishing momentain the small density limit, impurity lines coupled to thecurrent vertex, J x ↓ ( q ) = q x /M , are excluded. Thus, thecentral (red) line corresponds to a filled state. We mayset its momentum to zero as done in Fig. 3, and the in-tegration over filled states then simply produces a factorof n ↓ . J y ↑ ,αβ ( k ) J x ↓ ( q ) β ; ω k , k α ; Ω + ω k , k Ω + ω q , q ω q , q J y ↑ ,αβ ( k ) J x ↓ ( − q ) β ; ω k , k α ; Ω + ω k , k Ω + ω q , − q ω q , − q ω q + ˜ ω, ω k + ˜ ω, k − q ω q + ˜ ω,
1; Ω + ω k − ˜ ω, k − q Figure 3. Leading contributions to the drag transconduct-ivity. Dashed lines represent impurities, dotted lines interac-tion matrix elements W , see Eq. (10). The energy-momentumstructure of the central part and the colored elements are ex-plained in the main text. Identification of the red line with a filled state also fixesthe (red) index of the central majority line in order forthe ˜ ω integral (see Fig. 3) to be non-vanishing. Schemat-ically the top diagram in Fig. 3 describes the scatteringof an impurity with a particle, with momentum trans-fer q , and the bottom diagram the scattering with ahole, with momentum transfer − q . Therefore, the netmomentum transfer and drag vanish in the particle-holesymmetric case [22–24], as will be seen explictly below.The remaining evaluation of the diagrams is straightfor-ward (see App. B). We obtain σ ↓↑ = − g n ↓ (cid:90) d k (2 π ) d q (2 π ) Im (cid:110) J y ↑ , ( k ) W ( k − q , q ) W ( k , − q ) (cid:111) q x M (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k )) ( d ( k , q ) + c ( k , q )) , (12) d ( k , q ) = 2 (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k − q ) − (cid:15) ↓ ( q )( (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k − q ) − (cid:15) ↓ ( q )) , c ( k , q ) = 2 (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k − q ) + (cid:15) ↓ ( q )( (cid:15) ↑ , ( k − q ) − (cid:15) ↑ , ( k ) − (cid:15) ↓ ( q )) . (13)Here, c, d represent the contributions of the “direct” (topin Fig. 3) and “crossed” (bottom) diagrams. When flip-ping d → − d , we have (cid:15) → − (cid:15) and vice versa, thus σ ↓↑ is antisymmetric in d . In particular, it vanishes inthe particle-hole symmetric case, d = 0. Numerical eval-uation of Eq. (12) as function of d is shown in Fig. 4(a).Let us point out that the complete cancellation of σ ↓↑ at d = 0 only occurs to second order, O ( g ), and is notexpected in higher order, as can be shown explicitly forthe Haldane model (see below).In Fig. 4(b), σ ↓↑ is depicted as function of m for non-zero d , tuning the majority system from the trivial phasewith C = 0 to a non-trivial one, C = 1. While σ ↓↑ exhibitsa clear jump when the majority particles undergo a to-pological phase transition (see next section), it is neitherconstant in the non-trivial phase, nor does it vanish in thetrivial phase: For the majority particles, time-reversalsymmetry is broken everywhere in the phase diagram,but for C = 0 the transconductivity contributions of the“Dirac” and “spectator” fermions cancel exactly, as longas the chemical potential is in the gap and the lower ma-jority band is completely filled. In the case of the gaplessimpurity band, such cancellation is not guaranteed, andthe impurity Hall drag therefore does not vanish in thenon-trivial phase. (a)(b) d σ ↓↑ · (2 π ) g n ↓ σ ↓↑ · (2 π ) g n ↓ m C = 1 C = 0 Figure 4. Impurity transconductivity σ ↓↑ from numerical eval-uation of Eq. (12). Lines are guides for the eye. (a) σ ↓↑ asfunction of d for M = 1 , m = 0 . , d = −
1. (b) σ ↓↑ asfunction of m for M = 1 , d = − , d = 0 . III. The jump across the phase transition for the continuummodel
Another salient feature of Fig. 4(b) is the discontinu-ous change of the drag transconductivity which occursupon crossing the topological phase boundary m = 0.This jump can be understood as arising from a singularcontribution of Dirac fermions: When the gap closes, theDirac part of the majority Berry curvature ( ∝ m in Eq.(3)) evolves into a delta-function, sign( m ) δ (2) ( k ) – com- pare also Fig. 1(b). In contrast, the part correspondingto the spectator fermions ( ∝ d in Eq. (3)) is smoothacross the transition. In the expression for the impuritydrag (12), a singular Dirac contribution ∝ sign( m ) δ (2) ( k )arises as well. This singular contribution changes signacross the transition, and so induces the jump ∆ σ ↓↑ inthe Hall drag, with a sign determined by the change inChern number ∆ C . To extract ∆ σ ↓↑ we can set k = 0 inall parts of Eq. (12) which are non-singular as k →
0. Asdetailed in App. C, in this way we obtain σ ↓↑ , Dirac = (14) g n ↓ (cid:90) d k (2 π ) d q (2 π ) Im (cid:110) J y ↑ , Dirac , ( k ) J x ↑ , Dirac , ( k ) (cid:111) ( (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k )) q x M q (cid:112) d q ) (cid:18) (cid:15) ↓ ( q ) + (cid:15) ↑ , ( q )) − (cid:15) ↓ ( q ) − (cid:15) ↑ , ( q )) (cid:19) , where J x/y ↑ , Dirac ( k ) represents the majority current carriedby the Dirac (i.e., not the spectator) fermions. Comparedto Eq. (12) the k and q integrals in Eq. (14) have fac-torized. The k integral, which simplifies to an integralover a delta function as m →
0, is nothing but the Chernnumber contribution of the Dirac fermions, cf. Eq. (8).It evaluates to (1 / π )sgn( m ). Performing the remaining q integral, one finds σ ↓↑ , Dirac = − g n ↓ (2 π ) · π d M · sign( m )1 + 4 M ( | d | + ( d − d ) M ) . (15)Defining ∆ σ ↓↑ as the jump of Hall drag when goingfrom the trivial to the topological phase, with change inChern number ∆ C , Eq. (15) leads to the final result:∆ σ ↓↑ = ∆ C · g n ↓ (2 π ) (cid:18) − π d M M ( | d | + ( d − d ) M ) (cid:19) . (16)As a check, in Fig. 5 this formula is compared with a nu-merical evaluation of the jump from Eq. (12) as functionof the impurity mass M , yielding excellent agreement.Note that both Hall drag and jump will vanish in the lim-its M → M → ∞ : In the former limit, the impuritycannot interact efficiently with the majority particles dueto the large kinetic energy cost, while in the latter theimpurity is immobile and cannot be dragged.While the Dirac part of the Hall drag, σ ↓↑ , Dirac ,changes sign at the transition, there is also a smallsmooth background contribution from the spectator fer-mions, to be denoted σ ↓↑ , spec . This contribution can beextracted numerically from Eq. (12) as σ ↓↑ , spec = 12 (cid:2) σ ↓↑ ( m = 0 + ) + σ ↓↑ ( m = 0 − ) (cid:3) , (17)see the inset to Fig. 5. MMσ ↓↑ , spec · (2 π ) g n ↓ ∆ σ ↓↑ · (2 π ) g n ↓ Figure 5. Jump of the Hall drag ∆ σ ↓↑ in the continuum modelas function of M , with d = − , d = 0 .
5. The dashed linecorresponds to Eq. (16), points are computed numerically byevaluating Eq. (12) at two points m = ± .
001 close to thephase boundary. Numerical errors are of the order of thepoints size.
Inset : The smooth contribution of the spectatorfermions, obtained numerically from Eq. (12).
We note in passing that the jump of σ ↓↑ is reminis-cient of the recently shown [36] change of sign in the Hall coefficient for a single-particle gapless Dirac cone uponvariation of the particle density. IV. Drag and jump in the Haldane lattice model
The general behaviour of σ ↓↑ to leading order, O ( g ),is not limited to the continuum model (1), but will holdin other Chern insulators as well. As another example,we consider a situation [19] where the majority particlesare described by the Haldane model on the honeycomblattice [21], with Hamiltonian H ↑ ( k ) = (cid:88) i =0 ψ †↑ ( k ) ( h i ( k ) σ i ) ψ ↑ ( k ) , (18) ψ ↑ ( k ) = ( c ↑ ,A ( k ) , c ↑ ,B ( k )) T , k i = k · u i ,h ( k ) = − t (cid:48) cos( φ ) [cos( k − k ) + cos( k ) + cos( k )] ,h ( k ) = − [1 + cos( k ) + cos( k )] ,h ( k ) = − [(sin( k ) + sin( k )] ,h ( k ) =∆ / t (cid:48) sin( φ ) [sin( k − k ) + sin( k ) − sin( k )] , where u = (3 / , √ / T , u = (3 / , −√ / T , andthe lattice constant and nearest neighbour hopping amp-litude are set to 1. The reciprocal lattice vectors aregiven by b = (2 π/ , π/ √ T , b = ( − π/ , π/ √ T .The model is parametrized by the next-nearest-neighbourhopping t (cid:48) , the angle φ quantifying the time-reversal sym-metry breaking, and the sublattice potential offset ∆. Forgiven values of t (cid:48) , φ, ∆, the majority chemical potentialis implicitly placed in the gap (its precise value is irrelev-ant). The well-known topological phase diagram of theHaldane model is shown in Fig. 6(a).The impurity particles are governed by the tight-binding model for graphene (i.e., t (cid:48) = 0), with the chem-ical potential at the bottom of the lower band [19]: H ↓ ( k ) = (cid:88) i =0 ψ †↓ ( k ) h i ( k ) σ i ψ ↓ ( k ) , (19) ψ ↓ ( k ) = ( c ↓ ,A ( k ) , c ↓ ,B ( k )) T ,h ( k ) = 3 , h ( k ) = − [1 + cos( k · u ) + cos( k · u )] h ( k ) = − [(sin( k · u ) + sin( k · u )] , h ( k ) = 0 . The impurity-majority interaction, Eq. (10), is straigth-forwardly modified to account for the impurity multi-band structure – see App. D for details.The Hall drag σ ↓↑ can be derived in analogy to thecontinuum model, see Eq. (D2); the only minor change isthe appearance of diagonalizing unitary matrices U ↓ ( q )for the impurity. Numerical evaluation of σ ↓↑ is presen-ted in Figs. 6(b)–(d). Now, the particle-hole symmetriccase where (cid:15) = − (cid:15) corresponds to φ = ± π/
2, and σ ↓↑ vanishes accordingly [19]. Furthermore, one can easilydemonstrate the symmetry σ ↓↑ ( φ ) = − σ ↓↑ ( π − φ ), seeApp. D below Eq. (D2). This symmetry is readily seenin Fig. 6(c), which shows a cut through the phase dia-grams for fixed ∆ = 0. Combined with the symmetry σ ↓↑ ( φ ) = − σ ↓↑ ( − φ ) inherited from the Haldane model,this gives the Hall drag a periodicity σ ↓↑ ( φ ) = σ ↓↑ ( φ + π ) , (20)apparent in Fig. 6(b). This remarkable manifestationof particle-hole antisymmetry is in stark contrast to thepure majority case, where the Chern number only hasthe trivial periodicity C ( φ ) = C ( φ + 2 π ), see Fig. 6(a). At the special particle-hole symmetric parameterpoints, φ = ± π/ , ∆ = 0, one can also get insight intothe behavior of σ ↓↑ to higher order in g (see App. E):employing a particle-hole transformation which alsoexchanges band indices of the majority particles, itcan be shown that at these points the Hall drag isantisymmetric in g to all orders. So while there isno O ( g ) contribution, and the leading order, O ( g ),must vanish, at order O ( g ) the Hall drag will be nonzero.In the numerics, the jump of σ ↓↑ across the topolo-gical phase transition is again prominent, and clearlydelineates the topological phases of the parent Haldanemodel. Its origin is analogous to the continuum model– it comes from a sign-changing contribution of Diracfermions, which becomes singular upon gap closing. Theonly significant difference is that there are now two Diraccones in the problem, but except at the special points φ = 0 , π , the gap closes at only one of them. In the lan-guage employed for the continuum model, states near theDirac cone with open gap count as spectator fermions. Adetailed analysis of the jump leads to (see App. D)∆ σ ↓↑ = ∆ C · g n ↓ (2 π ) · f ( t (cid:48) , φ ) , (21)where f ( t (cid:48) , φ ) is a numerical function defined in Eq. (D4).It involves the remaining q integral, which is difficult toevaluate analytically in the lattice case. In Fig. 7(a),∆ σ ↓↑ is depicted as a function of φ . It is maximal as φ → + , π − , where the particle-hole asymmetry of thedispersion (away from the Dirac points) is largest. Again,the jump occurs on top of a smooth background con-tribution from the spectator fermions, presented in Fig.7(b). It too is maximal as φ → + , π − , approaching1 / σ ↓↑ : close to these angles, the spectator contribu-tion is almost fully determined by the second Dirac cone,which has a very small gap. Accordingly, the values ofthe sign-changing drag contribution, σ ↓↑ , Dirac , and thealmost Dirac-like background contribution are the same.
V. Measurement of the Hall drag via circular dichroism
The Hall drag σ ↓↑ has so far been related to a linearcurrent response to an external, linearly polarized elec-tric field, which is the standard point of view. However,recent theoretical works have shown [25–27, 37, 38] thattopological invariants can also be extracted from a meas-urement of excitation rates to second order in the amp-litudes of circularly polarized fields, which was verifiedin the experiment of Ref. [28]. For the Hall drag σ ↓↑ , arelation to an impurity excitation rate can be establishedas well, as we now show. Measuring such excitation ratesmay be a simpler route to detect σ ↓↑ experimentally, inparticular for ultracold gases.To set the stage, we first rephrase the results of Ref.[25] for the majority sector (non-interacting Chern insu-lator). The particles are coupled to external left or right π/ − π/ π − π − ∆ σ ↓↑ · (2 π ) g n ↓ σ ↓↑ · (2 π ) g n ↓ σ ↓↑ · (2 π ) g n ↓ ∆ / ∆ φφ = π/
4∆ = 0 φ ∆ C = 1 C = − − π π -∆ (a)(b) (c)(d) t = 0 . t = 0 . Figure 6. Impurity Hall drag σ ↓↑ in the Haldane model. (a) Majority phase diagram. ∆ = 6 √ t (cid:48) is the value of ∆ wherethe phase transition occurs for φ = π/
2. (b) σ ↓↑ from numerical evaluation of Eq. (D2) for t (cid:48) = 0 .
2. Cuts through the phasediagram along the dashed lines are shown in the next panels. (c) σ ↓↑ as function of φ for ∆ = 0 and two values of t (cid:48) . (d) σ ↓↑ as function of ∆ for φ = π/ t (cid:48) . The abscissa is rescaled by ∆ ( t (cid:48) ). circular polarized electrical fields: E ± ( t ) = 2 E (cos( ωt ) , ± sin( ωt )) T , (22)with ω a fixed drive frequency. In the temporal gauge,the time-dependent light-matter Hamiltonian reads H ↑ , ± ( t ) = 2 Eω (cid:16) ˆ J x ↑ sin( ωt ) ∓ ˆ J y ↑ cos( ωt ) (cid:17) . (23)When this perturbation is switched on, particles areexcited from the lower to the upper band. One candefine the associated depletion rates of initially occu-pied states with momentum k , Γ ↑ , ± ( k , ω ), which de-pend on the polarization of the driving field (“circu-lar dichroism”). In Ref. [25], these rates are obtainedfrom Fermi’s Golden Rule. Let ∆Γ ↑ ( ω ) be the differ-ence in total depletion rates for a fixed frequency ω ,∆Γ ↑ ( ω ) ≡ / (cid:80) k (Γ ↑ , + ( k , ω ) − Γ ↑ , − ( k , ω )). Then theChern number C follows the simple relation [39]: A E C = − (cid:90) ∞ dω ∆Γ ↑ ( ω ) . (24)This integration has to be understood as an average of∆Γ ↑ ( ω ) over different drive frequencies, obtained by re-peating the experiment many times [28].For our purposes here, it is useful to rederive the res-ult (24) from diagrammatic perturbation theory. This is achieved by relating the depletion rate to the on-shellretarded self-energy asΓ ± , ↑ ( k , ω ) = − ± ( (cid:15) ↑ , ( k ) , k ; ω )] . (25)In turn, the self-energy to second order in H ↑ , ± can berepresented by the Feynman diagram of Fig. 8, plus thediagram with the ˆ J x ↑ , ˆ J y ↑ vertices interchanged. The ne-cessary Feynman rules in energy-momentum space areeasily derived from H ↑ , ± , and are detailed in App. F.There are also processes involving ( ˆ J x ↑ ) , ( ˆ J y ↑ ) , but theycancel in ∆Γ ↑ ( ω ). Working directly in the real frequencyspace for convenience, ∆Γ ↑ ( ω ) can then be directly writ-ten down as∆Γ ↑ ( ω ) = − (cid:88) k Im [Σ + ( (cid:15) ( k ) , k ; ω ) − Σ − ( (cid:15) ( k ) , k ; ω )] = − (cid:88) k E ω Im (cid:20) (cid:16) iJ x ↑ , ( k ) J y ↑ , ( k ) − iJ y ↑ , ( k ) J x ↑ , ( k ) (cid:17) × ω + (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k ) + i + (cid:21) . (26) φφσ ↓↑ , spec · (2 π ) g n ↓ ∆ σ ↓↑ · (2 π ) g n ↓ (a)(b) Figure 7. (a) Jump of the Hall drag ∆ σ ↓↑ in the Haldanemodel as function of φ , with t (cid:48) = 0 . c tuned to thetransition line. The dashed line corresponds to formula (21),points are computed numerically by evaluating Eq. (D2) attwo points close to the phase boundary, with ∆ = ∆ c ± . c ± . σ ↓↑ ( φ = 0 + , π − ) / . Integrating over ω , we find: (cid:90) ∞ dω ∆Γ ↑ ( ω ) = (27)4 πE A (2 π ) (cid:90) d k (cid:90) ∞ dω δ ( ω − ( (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k ))) ω × Im (cid:104) J x ↑ , ( k ) J y ↑ , ( k ) (cid:105) (8) = − A E C , in agreement with Eq. (24). (cid:15) ( k ) , k (cid:15) ( k ) , k ω + (cid:15) ( k ) , k ω ωJ x ( k ) J y ( k ) Figure 8. On-shell self-energy diagram. Incoming and out-going fermion lines represent particles from the lower band,the intermediate line a particle from the upper band, andthe wiggly lines the circularly polarized electrical fields. TheFeynman rules are explained in App. F.
To summarize, we have related the majority Chernnumber to the differential depletion rate of filled statesfrom the lower band when the system is subjected to acircular perturbation. We can now extend this idea tothe impurity case. We consider our previous interactingmajority-impurity setup, with a small number of impurit-ies prepared in the lower band, and couple both majorityand impurity particles to the circular fields. On theirown, the impurities would not experience a differentialdepletion because of the time reversal invariance of theimpurity Hamiltonian. Only due the interaction with themajority particles such differential depletion will set in,corresponding to occupation of higher momentum states.Note that, for strong impurity-majority interactions, itwill rather be polaronic (dressed impurity) states whichare depleted. For weak coupling, however, such band-dressing effects can be neglected (to order O ( g )), andwe can think in terms of bare impurities in lieu of po-larons. In technical terms, our Feynman diagrams willnot contain any impurity self-energy insertions.Let us couple the impurities to the circular fields in thesame way as the majority particles, Eq. (23). We con-sider the depletion rate of the filled impurity state withvanishing momentum Γ ↓ , ± ( , ω ) ≡ Γ ↓ , ± ( ω ), which is ofmost interest when the impurity density is small. Sincenon-vanishing contributions to ∆Γ ↓ ( ω ) must involve ma-jority scattering, to order O ( g ) there are two classes ofrelevant diagrams; representative diagrams are shown inFig. 9.Consider first the two diagrams 9(a), 9(b) in the toprow of the Figure. These diagrams describe processeswhere only the majority particles are excited by the ex-ternal fields. Since they do not involve an impurity cur-rent, they are not related to the drag. Two additionaldiagrams where the direction of the external field lines isinverted can be drawn as well.The structural difference between Fig. 9(a) and 9(b)is the orientation of the majority lines, which maps toan inverted energy-momentum transfer on the impurity(marked red). Thus, similar to the drag diagrams of Fig.3, the diagrams are related by particle-hole symmetry.However, the contributions of these diagrams add uprather than cancel, since they do not contain an impuritycurrent operator, J ↓ ( q ), which is odd in q . Therefore, ascan be verified by a straightforward evaluation (cf. App.F, Eq. (F4)), the total contribution ∆Γ ↓ , ph of these dia-grams obeys ∆Γ ↓ , ph ( φ ) = ∆Γ ↓ , ph ( π − φ ) for the Haldaneand ∆Γ ↓ , ph ( d ) = ∆Γ ↓ , ph ( − d ) for the continuum model.As a result, in an experiment these processes can be pro-jected out by subtracting ∆Γ ↓ , ph ( φ ) − ∆Γ ↓ , ph ( π − φ ),which leaves out only the antisymmetric drag contribu-tion. Another way to separate ∆Γ ↓ , ph from the drag is tohave a different coupling constant between external fieldand impurities, which is feasible in the ultracold gas setupwhere the circular perturbation can for example be im-plemented by lattice shaking [28, 40]. Since ∆Γ ↓ , ph is in-dependent of the coupling to the impurities, it can againbe eliminated by subtracting measurements obtained for ω q , q ω q − ω, q , , J x ↓ ( q ) ωω ω + ω k − ω q , k − q ω k , k ω + ω k , k J y ↑ , ( k ) ω q − ω, q ω q , q , , J x ↓ ( q ) ωω ω + ω k − ω q , k − q ω + ω k , k ω k , k J y ↑ , ( k ) ω q , q , , ω + ω k − ω q , k − q ω k , k ω + ω k , k J x ↑ , ( k − q )1; ω k − ω q , k − q J y ↑ , ( k ) − ω q , − q , , ω + ω k − ω q , k − q ω k , k ω + ω k , k J x ↑ , ( k − q )1; ω k − ω q , k − q J y ↑ , ( k )(a) (b)(c) (d) Figure 9. Non-vanishing contributions to the impurity depletion rate Γ ↓ , ± ( , ω ). Panels (a), (b): Diagrams not related tothe drag, which are particle-hole symmetric. Panels (c), (d): Diagrams related to the drag. These two diagrams differ in theorientation of the field lines and the band index structure of the majority particles. two different impurity couplings.Let us assume either such elimination implicitly, andmove on to the two diagrams of Fig. 9(c), 9(d). In es-sence, they correspond to the drag transconductivity dia-gram of Fig. 3 (top), with the central (red) impurity linecut. The two other diagrams in this class have crossedinteraction lines, akin to the “crossed” diagrams of Fig. 3(bottom). Evaluation of these four diagrams is straight-forward, see App. F. Summation over the filled impuritystates simply yields: (cid:88) p , filled Γ ↓ , ± ( p , ω ) (cid:39) (cid:88) p , filled Γ ↓ , ± ( ω ) = A n ↓ Γ ↓ , ± ( ω ) . (28)For the integrated differential depletion rate, one thenfinds (cid:90) ∞ dω ∆Γ ↓ ,xy ( ω ) = 2 πA E σ ↓↑ , (29)as naively expected from Eq. (27). However, the impuritydepletion rate also receives contribution from processesinvolving the currents ˆ J y ↓ , ˆ J x ↑ . Per the Feynman rules (cf. App. F), these diagrams come with a relative minus sign,and then yield a factor of two for the total differentialrate, since σ xy, ↓↑ = − σ yx, ↓↑ for both the continuum andthe Haldane model, as one can check easily. Modulo theantisymmetrization discussed above, we therefore have σ ↓↑ = 14 πA E (cid:90) ∞ dω ∆Γ ↓ ( ω ) . (30)This result can also be rephrased in terms of excita-tion instead of depletion rates. Since the impurities areinitially prepared at the bottom of the band, one canwrite (cid:90) ∞ dω ∆Γ ↓ ( ω ) = (cid:88) q > (cid:90) ∞ dω ∆Γ ↓ , exc ( q , ω ) , (31)meaning that the impurities are excited into states withhigher momentum which are initially empty. These q -states correspond to the intermediate impurity lines inFig. 9. Via Eq. (30) we can then define a q -resolvedimpurity drag as σ ↓↑ ≡ (cid:88) q > σ ↓↑ ( q ) . (32)0This provides an alternative view on, say, the topo-logical jump ∆ σ ↓↑ . For the Haldane model, it can bephrased as ∆ σ ↓↑ = ∆ C (cid:82) d q f jump ( q ), where f jump ( q ) isa known function, see Eqs. (21), (D4). If the excitationrates defined in Eq. (31) can be experimentally detectedin q -resolved fashion (for instance with band mappingtechniques [41–43]), so can the q -resolved impurity drag σ ↓↑ ( q ). Measuring σ ↓↑ ( q ) at two points in the phase dia-gram close to the topological boundary then gives directaccess to f jump ( q ). Taken the other way around, suppos-ing that f jump ( q ) is known for the model realized in theexperiment, at each q -point an independent estimate ofthe change in Chern number across the phase transition∆ C is possible. VI. Conclusions
In this work we have studied to which extent a topolo-gically trivial impurity can be Hall-dragged by majorityexcitations in a Chern insulator, looking at two differ-ent models in a controlled perturbative setting. Sincethe impurity Hall drag is sensitive to the dispersion ofthe majority particles and holes, there is no one-to-onecorrespondence to the Chern number; nevertheless, thechange in Chern number across a topological transitionis clearly reflected by a discontinuous jump in the dragtransconductivity σ ↓↑ . Besides transport, this transcon-ductivity can also be extracted from a measurement ofimpurity excitation rates upon driving the system by acircularly polarized field, which is well suited for ultracoldgases.A worthwhile goal for future study is the extension tothe strong-coupling limit, in particular the analysis ofimpurity-majority bound state formation. These boundstates may have rather rich physics: they could inheritthe topological characteristics of the majority particles[15, 16], have opposite chirality as found for Haldanemodel in the two-body limit [44], or even be topologicalwhen the single-particle state are trivial [45–47]. Acknowledgments
We thank A. Kamenev for helpful discussions. D.P.acknowledges funding by the Deutsche Forschungsge-meinschaft (DFG, German Research Foundation) underGermany’s Excellence Strategy – EXC-2111 – 390814868,and is particularly grateful to the Max-Planck-Institutefor the Physics of Complex Systems Dresden (MPIPKS)for hospitality during the intermediate stage of thisproject. N.G. has been supported by the FRS-FNRS(Belgium) and the ERC Starting Grant TopoCold.P.M. has been supported by the Spanish MINECO (FIS2017-84114-C2-1- P), and EU FEDER Quantumcat.G.M.B. acknowledges support from the IndependentResearch Fund Denmark-Natural Sciences via Grant No.DFF-8021-00233B, and US Army CCDC Atlantic Basicand Applied Research via grant W911NF-19-1-0403.M.G. has been supported by the Israel Science Found-ation (Grant No. 227/15) and the US-Israel BinationalScience Foundation (Grant No. 2016224).
A. Basis rotation
The unitary matrix defined in Eq. (5) reads U ↑ ( k ) = (cid:18) U ↑ ,A ( k ) U ↑ ,A ( k ) U ↑ ,B ( k ) U ↑ ,B ( k ) (cid:19) , (A1) U ↑ ,A ( k ) = h ( k ) − h ( k ) (cid:112) h ( k )( h ( k ) − h ( k )) U ↑ ,A ( k ) = h ( k ) + h ( k ) (cid:112) h ( k )( h ( k ) + h ( k )) U ↑ ,B ( k ) = h ( k ) + ih ( k ) (cid:112) h ( k )( h ( k ) − h ( k )) ,U ↑ ,B ( k ) = h ( k ) + ih ( k ) (cid:112) h ( k )( h ( k ) + h ( k )) , where A, B refer to the sublattice- and 1 , J x/y ↑ = (cid:88) k c †↑ ,α ( k ) J x/y ↑ ,αβ ( k ) c ↑ ,β ( k ) , (A2)with matrix elements J x ↑ ( k ) = U †↑ ( k ) J x, ↑ ( k ) U ↑ ( k ) (A3) J x, ↑ ( k ) = ∂H ↑ ( k ) ∂k x = σ x + 2 k x ( d σ z + d ) , and likewise for J y ↑ ( k ). B. Evaluation of the drag diagrams in the continuum model
Let us start by considering the first “direct” diagramin Fig. 3 with majority band indices α = 1 , β = 2. Itscontribution to the Matsubara correlator − (cid:104) ˆ J x ↓ ˆ J y ↑ (cid:105) ( i Ω),to be denoted by P ( i Ω), reads1 P ( i Ω) = − g n ↓ A (cid:90) k,q (cid:90) d ˜ ω π J y ↑ , ( k ) W ( k − q , q ) W ( k , − q ) J x ↓ ( q ) (B1)1 i (Ω + ω q ) − (cid:15) ↓ ( q ) 1 iω q − (cid:15) ↓ ( q ) 1 i ( ω q + ˜ ω ) + 0 + i (Ω + ω k ) − (cid:15) ↑ , ( k ) 1 iω k − (cid:15) ↑ , ( k ) 1 i ( ω k + ˜ ω ) − (cid:15) ↑ , ( k − q ) , where 0 + in the third impurity propagator ensures the correspondence to filled states. Evaluating the frequencyintegrals we find: P ( i Ω) = − g n ↓ A (cid:90) d k (2 π ) d q (2 π ) J y ↑ , ( k ) W ( k − q , q ) W ( k , − q ) J x ↓ ( q ) (B2)1 (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k − q ) − (cid:15) ↓ ( q ) 1 − i Ω + (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k − q ) − (cid:15) ↓ ( q ) 1 − i Ω + (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k ) . Upon analytical continuation, i Ω → ω , only the O ( ω ) part contributes to the static drag as in the non-interactingcase. With Eq. (11), we get: σ ↓↑ , = ig n ↓ (cid:90) d k (2 π ) d q (2 π ) J y ↑ , ( k ) W ( k − q , q ) W ( k , − q ) q x M (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k )) d ( k , q ) , (B3)with d ( k , q ) as defined in Eq. (13). The remaining threecontributions to σ ↓↑ have the following structure: Thedirect diagram with majority indices α = 2 , β = 1 leadsto Eq. (B3) with A ≡ J y ↑ , ( k ) W ( k − q , q ) W ( k , − q )replaced by B ≡ − J y ↑ , ( k ) W ( k − q , q ) W ( k , − q );using elementary properties of unitary matrices, one canshow that B = − A (with A the complex conjugate of A ), thus yielding the part ∝ d ( k , q ) of Eq. (12) in themain text. The remaining “crossed” diagram of Fig. 3likewise generates the part ∝ c ( k , q ). C. Jump of the Hall drag in the continuum model
To derive the jump from Eq. (12), we need to projecton the part of the k -integrand corresponding to the Diracfermions, which becomes singular at k = 0 as m → k = 0 in all regular parts.The last factor in the integrand becomes d ( k , q ) + c ( k , q ) → (C1) (cid:18) (cid:15) ↓ ( q ) + (cid:15) ↑ , ( q )) − (cid:15) ↓ ( q ) − (cid:15) ↑ , ( q )) (cid:19) . In the part involving interaction matrices W , it is usefulto rewrite W ( k − q , q ) W ( k , − q ) (10) = (C2) U †↑ , n ( k ) U ↑ ,n ( k − q ) U †↑ , m ( k − q ) U ↑ ,m ( k ) → U †↑ , n ( k ) U ↑ ,n ( − q ) U †↑ , m ( − q ) U ↑ ,m ( k ) = (cid:16) U †↑ ( k ) V ( q ) U ↑ ( k ) (cid:17) ,V ( q ) nm ≡ U ↑ ,n ( − q ) U †↑ , m ( − q ) , where n, m are sublattice indices, and in the second stepwe have only kept the singular k dependence. V ( q ) isa hermitian matrix, and so can be expanded as a lin-ear combination of the unit and Pauli matrices with realcoefficients. Then it is easy to show that only the contri-bution ∝ σ x survives the integration in Eq. (12), whilethe rest either does not contribute to the required ima-ginary part or is antisymmetric in k x . Therefore, we canwrite (cid:16) U †↑ ( k ) V ( q ) U ↑ ( k ) (cid:17) ˆ= (C3) (cid:16) U †↑ ( k ) σ x U ↑ ( k ) (cid:17) Re [ V ( q ) ] = (cid:16) U †↑ ( k ) J x, ↑ , Dirac U ↑ ( k ) (cid:17) − q x q (cid:112) d q ) , where in the last step we identified the current operator ofthe Dirac fermions in the sublattice basis, σ x = J x, ↑ , Dirac (cf. Eq. (1)), and wrote out V ( q ) by inserting matrixelements of U ↑ ( − q ) from App. A. Inserting Eqs. (C1),(C3) into Eq. (12), we can write the sign-changing Diracpart of the Hall drag as shown in Eq. (14) of the maintext. D. Impurity Hall drag and jump in the Haldane model
In the Haldane model, we define the on-site interactionby (cf. Eq. (10))2 H int = gA (cid:88) (cid:96) = A,B (cid:88) k , p , q c †↑ ,(cid:96) ( k + q ) c ↑ ,(cid:96) ( k ) c †↓ ,(cid:96) ( p − q ) c ↓ ,(cid:96) ( p ) = gA (cid:88) k , p , q c †↑ ,α ( k + q ) c ↑ ,β ( k ) c †↓ , ( p − q ) c ↓ , ( p ) W αβ ( k , p , q ) ,W αβ ( k , p , q ) = (cid:88) (cid:96) = A,B U ↑ ,(cid:96)α ( k + q ) U ↑ ,(cid:96)β ( k ) U ↓ ,(cid:96) ( p − q ) U ↓ ,(cid:96) ( p ) . (D1)In Eq. (D1), we have restricted the impurity to the lower band, which is legitimate for weak interactions.With this interaction, the derivation of the Hall drag proceeds analogously to the continuum model, App. B, andresults in σ ↓↑ = (D2) − g n ↓ (cid:90) d k (2 π ) d q (2 π ) Im (cid:110) J y ↑ , ( k ) W ( k − q , q , q ) W ( k , , − q ) (cid:111) J x ↓ , ( q ) 1( (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k )) ( d ( k , q ) + c ( k , q )) , with J x ↓ , ( q ) the impurity current operator in the band basis (taking into account lower band contributions only),and c, d as in Eq. (13), only replacing the single band energy of the continuum model (cid:15) ↓ ( q ) by the lower band energy (cid:15) ↓ , ( q ).From Eq. (D2) one can readily derive the additional symmetry σ ↓↑ ( φ ) = − σ ↓↑ ( π − φ ) mentioned in the main text.In the majority Hamiltonian H ↑ ( k ), h ( k ; φ ) = − h ( k ; π − φ ), while the other coefficients are invariant under suchreflection. As a result, one finds c ( k , q ; φ ) = − d ( k , q ; π − φ ). All other elements of Eq. (D2) do not change, whichshows the property as claimed.To evaluate the jump of the Hall drag ∆ σ ↓↑ in the Haldane model in analogy with Sec. III, let us focus on thetransition line, ∆ c = 6 √ t (cid:48) sin( φ ), where the gap closes at the Dirac point k A = (0 , π/ √ T . Since σ ↓↑ is symmetricin ∆, for a given value of φ the value of ∆ σ ↓↑ at − ∆ c is the same. To extract the singular Dirac contribution at k A ,we let k → k A in all regular parts of Eq. (D2). In this limit, J x ↑ ( k ) → U †↑ ( k ) 32 σ y U ↑ ( k ) ≡ J x ↑ , Dirac ( k ) (D3)This current can be extracted from the interaction part of Eq. (D2) as in Sec. III, which allows to write the Diracpart of the Hall drag as σ ↓↑ , Dirac = g n ↓ (2 π ) (cid:90) d k π Im (cid:110) J y ↑ , Dirac , ( k ) J x ↑ , Dirac , ( k ) (cid:111) · (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k )) · f ( t (cid:48) , φ ) , (D4) f ( t (cid:48) , φ ) ≡ − π (cid:90) d q J x ↓ ( q ) (cid:32) (cid:15) ↑ , ( k A ) − (cid:15) ↑ , ( k A − q ) − (cid:15) ↓ , ( q )) − (cid:15) ↑ , ( k A − q ) − (cid:15) ↑ , ( k A ) − (cid:15) ↓ , ( q )) (cid:33) · Im (cid:110) U ↑ ,A ( k A − q ) U †↓ , A ( ) U ↓ ,A ( q ) U †↑ , B ( k A − q ) U ↓ ,B ( ) U †↓ , B ( q ) (cid:111) Again, the k and q integrals have factorized, and the k integral gives ± /
2. This yields a value of the jump as in Eq.(21) of the main text. The remaining q integral has to be evaluated numerically. E. Antisymmetry of the hall drag as function of g in theHaldane model with φ = ± π/ , ∆ = 0 . Here we show that the Hall drag σ ↓↑ in the Haldanemodel, with parameters φ = ± π/ , ∆ = 0, is antisym-metric in the impurity-majority coupling g to all or-ders. We work in the diagonal band frame, and performa particle-hole transformation which also exchanges theband indices: b ˜ α ( k ) ≡ c †↑ ,α ( − k ) , b † ˜ α ( k ) ≡ c ↑ ,α ( − k ) , ˜1 ≡ , ˜2 ≡ . (E1) Due to particle-hole symmetry for φ = ± π/
2, the formof the non-interacting majority Hamiltonian is invariantunder this transformation (up to a constant): H ↑ = (cid:88) k c †↑ ,α ( k ) (cid:15) α ( k ) c ↑ ,α ( k ) (E2)= (cid:88) k b † ˜ α ( − k ) [ − (cid:15) α ( k )] b ˜ α ( − k ) + const.= (cid:88) k b † α ( k ) (cid:15) α ( k ) b α ( k ) + const. , (cid:15) α ( k ) = − (cid:15) ˜ α ( − k ) was used. However, the interac-tion term acquires a minus sign under the variable trans-formation (E1): H int = gA (cid:88) k , p , q c †↑ ,α ( k + q ) c ↑ ,β ( k ) c †↓ , ( p − q ) c ↓ , ( p ) W αβ ( k , p , q ) (E3)= − gA (cid:88) k , p , q b † ˜ β ( − k ) b ˜ α ( − k − q ) c †↓ , ( p − q ) c ↓ , ( p ) W αβ ( k , p , q ) + const.= − gA (cid:88) k , p , q b † α ( k + q ) b β ( k ) c †↓ , ( p − q ) c ↓ , ( p ) W ˜ β ˜ α ( − k − q , p , q ) + const.= − gA (cid:88) k , p , q b † α ( k + q ) b β ( k ) c †↓ , ( p − q ) c ↓ , ( p ) W αβ ( k , p , q ) + const. . The unimportant additional terms are constant in themajority sector. In the last step, we used W ˜ β ˜ α ( − k − q , p , q ) = W αβ ( k , p , q ). This can be easily shown by inserting the matrix elements from Eqs. (A1), (18), butrequires h ( k ) = − h ( − k ), which is only fulfilled for ∆ =0 (and is violated in the continuum model).Last, the required majority current operator trans-forms as J y ↑ ,αβ ( k ) = (cid:88) k c †↑ ,α ( k ) U †↑ ,αn ( k ) (cid:2) J y, ↑ ( k ) (cid:3) nm U ↑ ,mβ ( k ) c ↑ ,β ( k ) , (cid:2) J y, ↑ ( k ) (cid:3) nm = (cid:2) ∂ k y H ↑ ( k ) (cid:3) nm , (E4) J y ↑ ,αβ ( k ) = − (cid:88) k b † ˜ β ( − k ) U †↑ ,αn ( k ) (cid:2) J y, ↑ ( k ) (cid:3) nm U ↑ ,mβ ( k ) b ˜ α ( − k ) + const.= − (cid:88) k b † α ( k ) U T ↑ , ˜ αm ( − k ) (cid:2) J y, ↑ ( − k ) (cid:3) Tmn U ↑ ,n ˜ β ( − k ) b β ( k ) + const. . Again, inserting matrix elements one can show that U T ↑ , ˜ αm ( − k ) (cid:2) J y, ↑ ( − k ) (cid:3) Tmn U ↑ ,n ˜ β ( − k ) = U †↑ ,αm ( k ) (cid:2) J y, ↑ ( k ) (cid:3) mn U ↑ ,nβ ( k ) , and the majority current changes sign. In conclusion, for φ = ± π/ , ∆ = 0 this proves the antisymmetry σ ↓↑ ( g ) = − σ ↓↑ ( − g ) , (E5)as claimed in the main text. F. σ ↓↑ from circular dichroism: Technical details The Feynman rules for the perturbation H ↑ , ± ( t ) of Eq.(23) in the energy-momentum domain are easily derivedfrom Wick’s theorem. They read: • Each current vertex comes with a factor
E/ω . • If an incoming (outgoing) electrical field linecouples to a J x -vertex, there is an extra factor − i ( i ) for both Γ ± ( ω ). • If an electrical field line (incoming or outgoing)couples to a J y -vertex, this gives a factor ∓ ± ( ω ).Application of these rules directly leads to Eq. (26) in thenon-interacting case. For the integrated impurity deple-tion rate, let us consider for instance the contribution ofthe two diagrams of Fig. 9(c), 9(d), to be denoted D . Itreads4 D = − n ↓ g E A (cid:90) ∞ dω (cid:90) d k (2 π ) d q (2 π ) Im (cid:26) (cid:90) dω k π (cid:90) dω q π (cid:16) − iJ y ↑ , ( k ) J x ↓ ( q ) W + 2 iJ y ↑ , ( k ) J x ↓ ( q ) W (cid:17) ω (F1)1 ω q − (cid:15) ↓ ( q ) + i + ω q − ω − (cid:15) ↓ ( q ) + i + ω k − (cid:15) ↑ , ( k ) − i + ω + ω k − (cid:15) ↑ , ( k ) + i + ω + ω k − ω q − (cid:15) ↑ , ( k − q ) + i + (cid:27) . Here W is shorthand for the proper interaction matrices (cf. Eq. (12)). The third propagator is advanced (it corres-ponds to a majority hole) and has a − i + term in the denominator, the other propagators are retarded. Performingthe ω k , ω q integrals yields D = − n ↓ g E A (cid:90) d k (2 π ) d q (2 π ) (cid:90) > dω Im (cid:26) (cid:16) − iJ y ↑ , ( k ) J x ↓ ( q ) W + 2 iJ y ↑ , ( k ) J x ↓ ( q ) W (cid:17) ω (F2)1 ω + (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k − q ) − (cid:15) ↓ ( q ) + i + (cid:15) ↑ , ( k ) − (cid:15) ↓ ( q ) − (cid:15) ↑ , ( k − q ) + i + ω + (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k ) + i + (cid:27) . The expression involving the currents is real, and the imaginary part comes from the propagators only. They yield asum of two delta-functions, since the propagator in the middle is real. Computing the ω -integral, after some trivialalgebra one then finds D = 2 πE A · − g n ↓ (cid:90) d k (2 π ) d q (2 π ) Im (cid:110) J y ↑ , ( k ) J x ↓ ( q ) W (cid:111) (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k − q ) − (cid:15) ↓ ( q )( (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k )) ( (cid:15) ↑ , ( k ) − (cid:15) ↑ , ( k − q ) − (cid:15) ↓ ( q )) , (F3)which is precisely 2 πE A times the σ ↓↑ -contribution of the “direct” diagram, cf. (12). Evaluation of the othernon-vanishing drag diagrams (crossed diagram and diagrams with J y ↓ , J x ↑ interchanged) proceeds in the same manner.Since diagrams where both external field lines couple to the impurity vanish when forming ∆Γ ↓ , the only remainingnon-zero diagrams are those of Fig. 9(a), 9(b) plus those with inverted directions of the external field lines. Aftersome straightforward simplifications, one finds a total contribution n ↓ g (2 π ) E A (cid:90) ∞ dωω (cid:90) d k d q Im (cid:104) J x ↑ , ( k − q ) J y ↑ , ( k ) W ( k , − q , − q ) W ( k − q , , q ) (cid:105) (F4)Im (cid:26) ( −
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