# Topological transport of mobile impurities

D. Pimenov<EMAIL_ADDRESS>William I. Fine Theoretical Physics Institute,
University of Minnesota, Minneapolis, MN 55455, USA  A. Camacho-Guardian
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.  N. Goldman  Université
Libre de Bruxelles, CP 231, Campus Plaine, 1050 Brussels, Belgium  P.
Massignan  Department de Física, Universitat Politècnica de Catalunya, Campus
Nord, B4-B5, E-08034 Barcelona, Spain  G. M. Bruun  Department of Physics and
Astronomy, Aarhus University, Ny Munkegade, DK-8000 Aarhus C, Denmark  M.
Goldstein  Raymond and Beverly Sackler School of Physics and Astronomy, Tel
Aviv University, Tel Aviv 6997801, Israel

###### Abstract

We study the Hall response of topologically-trivial mobile impurities (Fermi
polarons) interacting weakly with majority fermions forming a Chern-insulator
background. This setting involves a rich interplay between the genuine many-
body character of the polaron problem and the topological nature 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 compute the 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 Chern number characterizing the topological transport of the insulator on
its own – it also depends continuously on particle-hole breaking terms, to
which the Chern number is insensitive. However, when the insulator is tuned
across a topological phase transition, a sharp jump of the impurity Hall drag
results, for which we derive an analytical expression. We describe how to
experimentally detect the polaronic Hall drag and its characteristic jump,
setting the emphasis on the circular dichroism displayed by the impurity’s
absorption rate.

##  I Introduction

As a rule of thumb, interacting many-body systems in more than one dimension
are difficult to analyze, and controllable routes to the inclusion of
interactions are rare. One such approach is to consider a non-interacting
“majority” system, couple it to a small number of quantum impurities, and
study interaction effects on the impurities only. If the majority system is a
conventional metal, the impurities are transformed into so-called Fermi
polarons [1] 111Here, we understand the polaron as a mobile quasiparticle, and
not as a static impurity as recently studied in a topological system in Ref.
[55], which by now are routinely observed in ultracold-gas [3, 4, 5, 6, 7] and
also solid state experiments [8] – for a review, see for instance Refs. [9,
10, 11].

In these systems, the local kinematic properties of the impurities are
modified by the interaction with the medium, while the medium itself is
unmodified if the impurity density is small. The next logical question to ask
is whether global topological characteristics of the medium [12] can influence
the impurity as well: Can a topologically trivial impurity inherit the
topological quantum numbers of the medium? Such an interaction-induced
topology is a fundamentally interesting prospect. Furthermore, this question
is of high relevance to current cold-atom experiments, where a broad family of
topological band structures have been realized [13]. Topological and polaronic
physics are thus well-controlled (and highly active) but largely separate
fields in cold-atom research, and it is thus worthwhile and intriguing to
combine them together. This goal has been approached in a few recent
theoretical works, mainly from two perspectives: Either interaction effects
are strong such that an impurity-majority bound state is formed [14, 15, 16,
17], and the impurity inherits the topological quantum numbers of the
majority, or, alternatively, one can study the problem in weak coupling [18,
*PhysRevB.102.119903], as previously done by some of us, with the majority
forming a Chern insulator. This perturbative approach is well-controlled and
does not require additional regularization.

As a diagnostic tool for the inherited topological properties of the impurity
particles, Ref. [18, *PhysRevB.102.119903] numerically computed the impurity
Hall drag for majority particles governed by the Haldane lattice model [20].
It was found that the Hall drag is neither quantized nor simply follows the
majority phase diagram, and even vanishes in the center of the topological
phase; however, it exhibits a sharp jump upon tuning the insulator across its
topological phase transition. In this work, we introduce a generic (continuum)
Dirac model of a Chern insulator. This model follows the same universal
physics as the Haldane model, but allows for an analytical understanding of
the phenomena numerically observed in Ref. [18, *PhysRevB.102.119903].

With a diagrammatical approach, we show that the Hall drag can be split into
two drag contributions exerted by majority particles and holes, respectively.
These two contributions counteract each other, and completely cancel at the
particle-hole symmetric point. This is reminiscent of Coulomb drag in two-
layer systems [21, 22, 23], and explains the observed vanishing of the drag in
the center of the majority topological phase. If particle-hole symmetry is
broken, the impurity Hall drag can be non-vanishing even if the majority Chern
insulator is in the trivial phase. To understand the observed jump across the
topological phase transition, one should view the majority system as a
combination of Dirac-like fermions with linear dispersion, and “spectator”
fermions [24] with a quadratic dispersion. At the phase transition, the
spectator fermions change smoothly, but the Dirac fermions feel the gap
closing and exhibit a singular Berry curvature. We show that this singularity
is integrated over in the expression for the impurity Hall drag, which leads
to a jump proportional to the change in Chern number, including the correct
sign. This is the only clear manifestation of topology in weak-coupling
impurity transport. We derive an analytical formula for the jump, and validate
all results numerically for the Haldane lattice model.

To supplement the theoretical results, we present a detailed discussion on how
to detect the Hall drag and jump with various experimental techniques. A
particular promising approach is to use circular dichroism, that is, measuring
impurity excitation rates upon driving the system with left and right
circularly polarized fields [25, 26, 27, 28]. A systematic method of computing
the excitation rates in an interacting many-body system is presented along the
way.

The remainder of this paper is structured as follows: In Sec. II we present
the continuum Dirac model and the evaluation of the impurity drag. In Sec.
III, we investigate the jump across the topological phase transition. The drag
including its jump at the topological transition is analyzed for the Haldane
model in Sec. IV. The different measurement protocols are detailed in Sec. V,
with special focus on the dichroic measurement. Conclusions and outlook are
presented in Sec. Acknowledgments. Some technical details are relegated to
Appendices.

##  II Drag transconductivity in the continuum model

We start by computing the impurity drag in a generic continuum model and
consider the following two-dimensional Bloch Hamiltonian for majority
particles indexed by a pseudospin $\uparrow$:

$\displaystyle
H_{\uparrow}({\boldsymbol{k}})=\sum_{i=0}^{3}\psi_{\uparrow}^{\dagger}({\boldsymbol{k}})h_{i}({\boldsymbol{k}})\sigma_{i}\psi_{\uparrow}({\boldsymbol{k}})\
,$ (1)
$\displaystyle\psi_{\uparrow}({\boldsymbol{k}})=\left(c_{\uparrow,A}({\boldsymbol{k}}),c_{\uparrow,B}({\boldsymbol{k}})\right)^{T}\
,$ $\displaystyle h_{1}({\boldsymbol{k}})=k_{x}\ ,\quad
h_{2}({\boldsymbol{k}})=k_{y}\ ,\quad h_{3}({\boldsymbol{k}})=m+d_{1}k^{2},$
$\displaystyle h_{0}({\boldsymbol{k}})=d_{2}k^{2},\quad k=|{\boldsymbol{k}}|\
,$

with $\sigma_{0}=\mathbbm{1}$ and $\sigma_{i}$ with $i=1,2,3$ being the Pauli
matrices. Throughout this paper we will work in units where $\hbar=c=e=1$; all
quantities are measured in appropriate powers of the (inverse) physical
fermion mass, while momenta are rescaled by the band velocity. Equation (1)
can be seen as a low-energy approximation to a microscopic tight-binding
Hamiltonian with a two-sublattice structure ($A,B$) and broken time-reversal
invariance. The eigenenergies corresponding to (1) read

$\displaystyle\epsilon_{\uparrow;1,2}({\boldsymbol{k}})=h_{0}({\boldsymbol{k}})\mp
h({\boldsymbol{k}}),\quad h({\boldsymbol{k}})=\sqrt{k^{2}+h_{3}(k)^{2}}\ .$
(2)

Without the terms $d_{1},d_{2}$ (which have physical dimensions (mass)-1), Eq.
(1) describes a gapped Dirac cone with mass gap $m$. The term $d_{1}$ serves
as a UV regularizer and makes the dispersion quadratic at higher energies
while preserving particle-hole symmetry,
$\epsilon_{\uparrow,1}({\boldsymbol{k}})=-\epsilon_{\uparrow,2}({\boldsymbol{k}})$.
The symmetry is broken for finite $d_{2}$. We assume $|d_{1}|>|d_{2}|$, thus
the lower (upper) band is filled (empty).

For general $d_{2}$, the Hamiltonian (1) is in the Altland-Zirnbauer class A
[29], and gives rise to a quantized Chern number $\mathcal{C}$. As shown
below, it reads

$\displaystyle\mathcal{C}$
$\displaystyle=\frac{1}{2\pi}\int\\!d{\boldsymbol{k}}\frac{1}{2}\frac{(m-d_{1}k^{2})}{(k^{2}+(m+d_{1}k^{2})^{2})^{3/2}}$
(3) $\displaystyle=\frac{1}{2}\left[\text{sign}(m)-\text{sign}(d_{1})\right]\
.$

The integrand of Eq. (3) is nothing but the Berry curvature
$\mathcal{F}_{xy}(k)$. As visualized in Fig. 1, for $m\rightarrow 0$
$\mathcal{F}_{xy}(k)$ consists of a sharp half-quantized peak for $k\lesssim
m$, arising from the Dirac fermions, on top of a broad background from high-
energy “spectator” fermions [24]. Both types of fermions effectively
contribute a half-integer Chern number, such that the total Chern number is
quantized to an integer.

Figure 1: Berry curvature for $d_{1}=-1$ and $m=\pm 0.1$ (full lines), $m=\pm
0.2$ (dashed lines). The inset shows a zoom-in on small values of
$\mathcal{F}_{xy}(k)$, highlighting the sign-change of the Berry curvature in
the trivial phase.

As explicit in Eq. (3), $\mathcal{C}$ does not depend on the particle-hole
symmetry breaking parameter $d_{2}$. This is in line with the geometrical
interpretation of $\mathcal{C}$ as a winding number 222For the winding number
construction one should view momentum space as compactified,
$\mathbb{R}_{2}\rightarrow S_{2}$, which is independent of the term $h_{0}$
commuting with the Hamiltonian [31].

As a preparation for the later calculations, it is useful to recap the
computation of $\mathcal{C}$ explicitly as $\mathcal{C}=-2\pi{\sigma}_{xy}$
[32, 33], with $\sigma_{xy}$ the transconductivity; the conductivity quantum
is $\sigma_{0}=e^{2}/\hbar=1/2\pi$ with the chosen units. In linear response,
$\sigma_{xy}$ is proportional to the retarded current-current correlation
function, which may be obtained by analytical continuation from imaginary
time:

$\displaystyle\sigma_{xy}=\lim_{\omega\rightarrow 0}\frac{1}{-i\omega
A_{0}}\left[-\braket{\hat{J}_{\uparrow}^{x}\hat{J}_{\uparrow}^{y}}(i\Omega)\bigg{|}_{i\Omega\rightarrow\omega+i0^{+}}\right],$
(4)

with $A_{0}$ the system area, and $\hat{J}_{\uparrow}$ the current operators
at vanishing external momentum.

The imaginary time correlator in Eq. (4) can be written as

$\displaystyle-\braket{\hat{J}_{\uparrow}^{x}\hat{J}_{\uparrow}^{y}}(i\Omega)=$
(5) $\displaystyle
A_{0}\int_{k}G_{\uparrow,\alpha}(\omega_{k},{\boldsymbol{k}})G_{\uparrow,\beta}(\Omega+\omega_{k},{\boldsymbol{k}})J^{x}_{\uparrow,\alpha\beta}({\boldsymbol{k}})J^{y}_{\uparrow,\beta\alpha}({\boldsymbol{k}}),$
$\displaystyle\int_{k}\equiv\int\frac{d{\boldsymbol{k}}d\omega_{k}}{(2\pi)^{3}},\quad
G_{\uparrow,\alpha}(\omega_{k},{\boldsymbol{k}})=\frac{1}{i\omega_{k}-\epsilon_{\uparrow,\alpha}({\boldsymbol{k}})}\
,$

where $\alpha,\beta$ refer to band indices and the Einstein summation
convention is implied. $J_{\uparrow,\alpha\beta}^{x/y}$ are current matrix
element in the diagonal basis (see App. A for details). The standard
diagrammatical representation of Eq. (5) is shown in Fig. 2. The Matsubara
Green function $G_{\uparrow,1}$ describes the propagation of a hole in the
filled lower band, while $G_{\uparrow,2}$ represents a particle in the upper
band. The frequency integral in Eq. (5) only receives contributions when
$\alpha\neq\beta$, and thus one can view creation of virtual particle-hole
pairs as the origin of the conductivity. These quasiparticles are virtual,
since the external field does not provide enough energy ($\Omega\rightarrow
0$) to overcome the band gap.

Figure 2: Diagram representing Eq. (5), with $\alpha=1,\beta=2$.

Evaluation of Eqs. (5) and (4) is straightforward. One finds

$\displaystyle\sigma_{xy}$
$\displaystyle=-i\int\\!\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}\frac{J_{\uparrow,12}^{x}({\boldsymbol{k}})J^{y}_{\uparrow,21}({\boldsymbol{k}})-J_{\uparrow,21}^{x}({\boldsymbol{k}})J^{y}_{\uparrow,12}({\boldsymbol{k}})}{(\epsilon_{\uparrow,1}({\boldsymbol{k}})-\epsilon_{\uparrow,2}({\boldsymbol{k}}))^{2}}$
$\displaystyle=-\frac{1}{2\pi}\mathcal{C}\ .$ (6)

Inserting current matrix elements and dispersions into Eq. (6) produces Eq.
(3). After this noninteracting prelude, we are ready to attack the polaron
problem. We consider a minority particle species indexed by $\downarrow$, with
a trivial quadratic Hamiltonian $H_{\downarrow}({\boldsymbol{p}})$:

$\displaystyle
H_{\downarrow}({\boldsymbol{p}})=\epsilon_{\downarrow}({\boldsymbol{p}})c^{\dagger}_{\downarrow}({\boldsymbol{p}})c_{\downarrow}({\boldsymbol{p}}),\quad\epsilon_{\downarrow}({\boldsymbol{p}})=\frac{p^{2}}{2M}\
.$ (7)

We can view the impurities as governed by a similar tight-binding Hamiltonian
as the majority, but with a chemical potential almost at the bottom of the
lower band, around which the dispersion is approximated by an effective mass
$M$. Higher impurity bands can be safely neglected.

The majority and minority particles interact via an onsite-interaction
$H_{\text{int}}$ [18, *PhysRevB.102.119903], which does not distinguish
between the sublattices (recall that the sublattices give rise to the two-band
structure):

$\displaystyle
H_{\text{int}}=\frac{g}{A_{0}}\sum_{\ell=A,B}\sum_{{\boldsymbol{k}},{\boldsymbol{p}},{\boldsymbol{q}}}c^{\dagger}_{\uparrow,\ell}({\boldsymbol{k}}+{\boldsymbol{q}})c_{\uparrow,\ell}({\boldsymbol{k}})c^{\dagger}_{\downarrow}({\boldsymbol{p}}-{\boldsymbol{q}})c_{\downarrow}({\boldsymbol{p}})=$
$\displaystyle\frac{g}{A_{0}}\sum_{{\boldsymbol{k}},{\boldsymbol{p}},{\boldsymbol{q}}}c^{\dagger}_{\uparrow,\alpha}({\boldsymbol{k}}+{\boldsymbol{q}})c_{\uparrow,\beta}({\boldsymbol{k}})c^{\dagger}_{\downarrow}({\boldsymbol{p}}-{\boldsymbol{q}})c_{\downarrow}({\boldsymbol{p}})W_{\alpha\beta}({\boldsymbol{k}},{\boldsymbol{q}}),$
$\displaystyle
W_{\alpha\beta}({\boldsymbol{k}},{\boldsymbol{q}})\equiv\left[U_{\uparrow}^{\dagger}({\boldsymbol{k}}+{\boldsymbol{q}})U_{\uparrow}({\boldsymbol{k}})\right]_{\alpha\beta}\
,$ (8)

where we have rotated to the band space in the second line. Now we imagine a
constant and uniform force $\boldsymbol{E}=E\textbf{e}_{y}$ acting on both
majority and minority particles 333Note that $e=1$ is the effective charge
corresponding to this force and might not be directly related to the electron
charge. Due to the interaction $H_{\text{int}}$, a transverse impurity current
$J_{\downarrow}^{x}$ will be induced; without interaction, there is none due
to time reversal symmetry of the impurities. To quantify this effect, we must
compute the Hall drag transconductivity

$\displaystyle\sigma_{\downarrow\uparrow}\equiv\lim_{\omega\rightarrow
0}\frac{1}{-i\omega
A_{0}}\left[-\braket{\hat{J}_{\downarrow}^{x}\hat{J}_{\uparrow}^{y}}(i\Omega)\bigg{|}_{i\Omega\rightarrow\omega+i0^{+}}\right]\
.$ (9)

This computation will be done to second order in the impurity-majority
coupling $g$, since the first order contribution vanishes [18,
*PhysRevB.102.119903]; thus, attractive and repulsive interactions lead to the
same result. We point out that such perturbative expansion is well-controlled
for small $g$, and no resummation is needed, in contrast with the recent
evaluation of longitudinal polaron drag in the metallic case [35].

Figure 3: Leading contributions to the drag transconductivity. Dashed lines
represent impurities, dotted lines interaction matrix elements $W$, see Eq.
(8). The energy-momentum structure of the central part and the colored
elements are explained in the main text.

As in the case of Coulomb drag in two-layer systems [21], the
$\mathcal{O}(g^{2})$ contribution corresponds to the two diagrams shown in
Fig. 3. We evaluate these diagrams to leading order in the small impurity
density $n_{\downarrow}$. The diagrams involve an impurity loop and are
therefore proportional to $n_{\downarrow}$, unlike the single-particle polaron
diagrams which have an impurity “backbone” [36, *PhysRevB.77.020408]. It is
convenient to identify the impurity lines that represent filled states
($\hat{=}$ impurity holes). Since these carry vanishing momenta in the small
density limit, impurity lines coupled to the current vertex,
$J_{\downarrow}^{x}({\boldsymbol{q}})=q_{x}/M$, are excluded. Thus, the
central (red) line corresponds to a filled state. We may set its momentum to
zero as done in Fig. 3, and the integration over filled states then simply
produces a factor of $n_{\downarrow}$.

Identification of the red line with a filled state also fixes the (red) index
of the central majority line in order for the $\tilde{\omega}$ integral (see
Fig. 3) to be non-vanishing. Schematically the top diagram in Fig. 3 describes
the scattering of an impurity with a particle, with momentum transfer
${\boldsymbol{q}}$, and the bottom diagram the scattering with a hole, with
momentum transfer $-{\boldsymbol{q}}$. Therefore, the net momentum transfer
and drag vanish in the particle-hole symmetric case [21, 22, 23], as will be
seen explictly below. The remaining evaluation of the diagrams is
straightforward (see App. B). We obtain

$\displaystyle\sigma_{\downarrow\uparrow}=-2g^{2}n_{\downarrow}\int\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}\frac{d{\boldsymbol{q}}}{(2\pi)^{2}}\
\text{Im}\left\\{J^{y}_{\uparrow,12}({\boldsymbol{k}})W^{22}({\boldsymbol{k}}-{\boldsymbol{q}},{\boldsymbol{q}})W^{21}({\boldsymbol{k}},-{\boldsymbol{q}})\right\\}\frac{q_{x}}{M}\frac{1}{\left({\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}})\right)^{2}}\left(d({\boldsymbol{k}},{\boldsymbol{q}})+c({\boldsymbol{k}},{\boldsymbol{q}})\right),$
(10) $\displaystyle
d({\boldsymbol{k}},{\boldsymbol{q}})=\frac{2\epsilon_{\uparrow,1}({\boldsymbol{k}})-\epsilon_{\uparrow,2}({\boldsymbol{k}})-\epsilon_{\uparrow,2}({\boldsymbol{k}}-{\boldsymbol{q}})-\epsilon_{\downarrow}({\boldsymbol{q}})}{\left({\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}}-{\boldsymbol{q}})-\epsilon_{\downarrow}({\boldsymbol{q}})\right)^{3}}\
,\quad
c({\boldsymbol{k}},{\boldsymbol{q}})=\frac{2{\epsilon_{\uparrow,2}}({\boldsymbol{k}})-{\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\uparrow,1}}({\boldsymbol{k}}-{\boldsymbol{q}})+\epsilon_{\downarrow}({\boldsymbol{q}})}{({\epsilon_{\uparrow,1}}({\boldsymbol{k}}-{\boldsymbol{q}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}})-\epsilon_{\downarrow}({\boldsymbol{q}}))^{3}}\
.$ (11)

Here, $c,d$ represent the contributions of the “direct” (top in Fig. 3) and
“crossed” (bottom) diagrams. When flipping $d_{2}\rightarrow-d_{2}$, we have
$\epsilon_{1}\rightarrow-\epsilon_{2}$ and vice versa, thus
${\sigma_{\downarrow\uparrow}}$ is antisymmetric in $d_{2}$. In particular, it
vanishes in the particle-hole symmetric case, $d_{2}=0$. Numerical evaluation
of Eq. (10) as function of $d_{2}$ is shown in Fig. 4(a). Let us point out
that the complete cancellation of $\sigma_{\downarrow\uparrow}$ at $d_{2}=0$
only occurs to second order, $\mathcal{O}(g^{2})$, and is not expected in
higher order, as can be shown explicitly for the Haldane model (see below).

In Fig. 4(b), ${\sigma_{\downarrow\uparrow}}$ is depicted as function of $m$
for non-zero $d_{2}$, tuning the majority system from the trivial phase with
$\mathcal{C}=0$ to a non-trivial one, $\mathcal{C}=1$. While
${\sigma_{\downarrow\uparrow}}$ exhibits a clear jump when the majority
particles undergo a topological phase transition (see next section), it is
neither constant in the non-trivial phase, nor does it vanish in the trivial
phase: For the majority particles, time-reversal symmetry is broken everywhere
in the phase diagram, but for $\mathcal{C}=0$ the transconductivity
contributions of the “Dirac” and “spectator” fermions cancel exactly, as long
as the chemical potential is in the gap and the lower majority band is
completely filled. In the case of the gapless impurity band, such cancellation
is not guaranteed, and the impurity Hall drag therefore does not vanish in the
non-trivial phase.

Figure 4: Impurity transconductivity $\sigma_{\downarrow\uparrow}$ from
numerical evaluation of Eq. (10). Lines are guides for the eye. (a)
$\sigma_{\downarrow\uparrow}$ as function of $d_{2}$ for $M=1,m=0.2,d_{1}=-1$.
(b) $\sigma_{\downarrow\uparrow}$ as function of $m$ for
$M=1,d_{1}=-1,d_{2}=0.5$.

##  III The jump across the phase transition for the continuum model

Another salient feature of Fig. 4(b) is the discontinuous change of the drag
transconductivity which occurs upon crossing the topological phase boundary
$m=0$. This jump can be understood as arising from a singular contribution of
Dirac fermions: When the gap closes, the Dirac part of the majority Berry
curvature ($\propto m$ in Eq. (3)) evolves into a delta-function,
$\text{sign}(m)\delta^{(2)}({\boldsymbol{k}})$ – compare also Fig. 1. In
contrast, the part corresponding to the spectator fermions ($\propto d_{1}$ in
Eq. (3)) is smooth across the transition. In the expression for the impurity
drag (10), a singular Dirac contribution
$\propto\text{sign}(m)\delta^{(2)}({\boldsymbol{k}})$ arises as well. This
singular contribution changes sign across the transition, and so induces the
jump $\Delta{\sigma_{\downarrow\uparrow}}$ in the Hall drag, with a sign
determined by the change in Chern number $\Delta\mathcal{C}$. To extract
$\Delta{\sigma_{\downarrow\uparrow}}$ we can set ${\boldsymbol{k}}=0$ in all
parts of Eq. (10) which are non-singular as ${\boldsymbol{k}}\rightarrow 0$.
As detailed in App. C, in this way we obtain

$\displaystyle\sigma_{\downarrow\uparrow,\text{Dirac}}=$ (12) $\displaystyle
g^{2}n_{\downarrow}\int\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}\frac{d{\boldsymbol{q}}}{(2\pi)^{2}}\
\frac{\text{Im}\left\\{J^{y}_{\uparrow,\text{Dirac},12}({\boldsymbol{k}})J^{x}_{\uparrow,\text{Dirac},21}({\boldsymbol{k}})\right\\}}{\left({\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}})\right)^{2}}\frac{q_{x}^{2}}{Mq\sqrt{1+(d_{1}q)^{2}}}\left(\frac{1}{(\epsilon_{\downarrow}({\boldsymbol{q}})+{\epsilon_{\uparrow,2}}({\boldsymbol{q}}))^{2}}-\frac{1}{(\epsilon_{\downarrow}({\boldsymbol{q}})-{\epsilon_{\uparrow,1}}({\boldsymbol{q}}))^{2}}\right)\
,$

where $J^{x/y}_{\uparrow,\text{Dirac}}({\boldsymbol{k}})$ represents the
majority current carried by the Dirac (i.e., not the spectator) fermions.
Compared to Eq. (10) the ${\boldsymbol{k}}$ and ${\boldsymbol{q}}$ integrals
in Eq. (12) have factorized. The ${\boldsymbol{k}}$ integral, which simplifies
to an integral over a delta function as $m\rightarrow 0$, is nothing but the
Chern number contribution of the Dirac fermions, cf. Eq. (6). It evaluates to
$(1/8\pi)\text{sgn}(m)$. Performing the remaining ${\boldsymbol{q}}$ integral,
one finds

$\displaystyle\sigma_{\downarrow\uparrow,\text{Dirac}}=-\frac{g^{2}n_{\downarrow}}{(2\pi)^{4}}\cdot\frac{4\pi^{2}d_{2}M\cdot\text{sign}(m)}{1+4M(|d_{1}|+(d_{1}^{2}-d_{2}^{2})M)}\
.$ (13)

Defining $\Delta{\sigma_{\downarrow\uparrow}}$ as the jump of Hall drag when
going from the trivial to the topological phase, with change in Chern number
$\Delta\mathcal{C}$, Eq. (13) leads to the final result:

$\displaystyle\Delta{\sigma_{\downarrow\uparrow}}=\Delta\mathcal{C}\cdot\frac{g^{2}n_{\downarrow}}{(2\pi)^{4}}\left(-\frac{8\pi^{2}d_{2}M}{1+4M(|d_{1}|+(d_{1}^{2}-d_{2}^{2})M)}\right)\
.$ (14)

As a check, in Fig. 5 this formula is compared with a numerical evaluation of
the jump from Eq. (10) as function of the impurity mass $M$, yielding
excellent agreement. Note that both Hall drag and jump will vanish in the
limits $M\to 0\ \text{or}\ M\to\infty$: In the former limit, the impurity
cannot interact efficiently with the majority particles due to the large
kinetic energy cost, while in the latter the impurity is immobile and cannot
be dragged.

Figure 5: Jump of the Hall drag $\Delta{\sigma_{\downarrow\uparrow}}$ in the
continuum model as function of $M$, with $d_{1}=-1,d_{2}=0.5$. The dashed line
corresponds to Eq. (14), points are computed numerically by evaluating Eq.
(10) at two points $m=\pm 0.001$ close to the phase boundary. Numerical errors
are of the order of the points size. Inset: The smooth contribution of the
spectator fermions, obtained numerically from Eq. (10).

While the Dirac part of the Hall drag,
$\sigma_{\downarrow\uparrow,\text{Dirac}}$, changes sign at the transition,
there is also a small smooth background contribution from the spectator
fermions, to be denoted $\sigma_{\downarrow\uparrow,\text{spec}}$. This
contribution can be extracted numerically from Eq. (10) as

$\displaystyle\sigma_{\downarrow\uparrow,\text{spec}}=\frac{1}{2}\left[{\sigma_{\downarrow\uparrow}}(m=0^{+})+{\sigma_{\downarrow\uparrow}}(m=0^{-})\right]\
,$ (15)

see the inset to Fig. 5.

We note in passing that the jump of ${\sigma_{\downarrow\uparrow}}$ is
reminiscient of the recently shown [38] change of sign in the Hall coefficient
for a single-particle gapless Dirac cone upon variation of the particle
density.

##  IV Drag and jump in the Haldane lattice model

The general behaviour of $\sigma_{\downarrow\uparrow}$ to leading order,
$\mathcal{O}(g^{2})$, is not limited to the continuum model (1), but will hold
in other Chern insulators as well. As another example, we consider a situation
[18, *PhysRevB.102.119903] where the majority particles are described by the
Haldane model on the honeycomb lattice [20], with Hamiltonian

$\displaystyle H_{\uparrow}({\boldsymbol{k}})$
$\displaystyle=\sum_{i=0}^{3}\psi_{\uparrow}^{\dagger}({\boldsymbol{k}})\left(h_{i}({\boldsymbol{k}})\sigma_{i}\right)\psi_{\uparrow}({\boldsymbol{k}})\
,$ (16) $\displaystyle\psi_{\uparrow}({\boldsymbol{k}})$
$\displaystyle=\left(c_{\uparrow,A}({\boldsymbol{k}}),c_{\uparrow,B}({\boldsymbol{k}})\right)^{T},\
\quad{\boldsymbol{k}}_{i}={\boldsymbol{k}}\cdot\boldsymbol{u}_{i}\ ,$
$\displaystyle h_{0}({\boldsymbol{k}})$
$\displaystyle=-2t^{\prime}\cos(\phi)\left[\cos({\boldsymbol{k}}_{1}-{\boldsymbol{k}}_{2})+\cos({\boldsymbol{k}}_{1})+\cos({\boldsymbol{k}}_{2})\right]\
,$ $\displaystyle h_{1}({\boldsymbol{k}})$
$\displaystyle=-\left[1+\cos({\boldsymbol{k}}_{1})+\cos({\boldsymbol{k}}_{2})\right]\
,$ $\displaystyle h_{2}({\boldsymbol{k}})$
$\displaystyle=-\left[(\sin({\boldsymbol{k}}_{1})+\sin({\boldsymbol{k}}_{2})\right],$
$\displaystyle h_{3}({\boldsymbol{k}})$ $\displaystyle=$
$\displaystyle\Delta/2+2t^{\prime}\sin(\phi)\left[\sin({\boldsymbol{k}}_{1}-{\boldsymbol{k}}_{2})+\sin({\boldsymbol{k}}_{2})-\sin({\boldsymbol{k}}_{1})\right]\
,$

where $\ {\boldsymbol{u}_{1}}=(3/2,\sqrt{3}/2)^{T},\
{\boldsymbol{u}_{2}}=(3/2,-\sqrt{3}/2)^{T}$, and the lattice constant and
nearest neighbour hopping amplitude are set to 1. The reciprocal lattice
vectors are given by $\boldsymbol{b}_{1}=(2\pi/3,2\pi/\sqrt{3})^{T},\
\boldsymbol{b}_{2}=(-2\pi/3,2\pi/\sqrt{3})^{T}$. The model is parametrized by
the next-nearest-neighbour hopping $t^{\prime}$, the angle $\phi$ quantifying
the time-reversal symmetry breaking, and the sublattice potential offset
$\Delta$. For given values of $t^{\prime},\phi,\Delta$, the majority chemical
potential is implicitly placed in the gap (its precise value is irrelevant).
The well-known topological phase diagram of the Haldane model is shown in Fig.
6(a).

The impurity particles are governed by the tight-binding model for graphene
(i.e., $t^{\prime}=\Delta=0$), with the chemical potential at the bottom of
the lower band [18] by setting $h_{0}({\boldsymbol{k}})=3$. The impurity-
majority interaction, Eq. (8), is straigthforwardly modified to account for
the impurity multi-band structure.

The Hall drag ${\sigma_{\downarrow\uparrow}}$ can then be derived in analogy
to the continuum model, see App. D for details; the only minor change is the
appearance of diagonalizing unitary matrices
$U_{\downarrow}({\boldsymbol{q}})$ for the impurity. Numerical evaluation of
${\sigma_{\downarrow\uparrow}}$ is presented in Figs. 6(b)–(d). Now, the
particle-hole symmetric case where $\epsilon_{1}=-\epsilon_{2}$ corresponds to
$\phi=\pm\pi/2$, and ${\sigma_{\downarrow\uparrow}}$ vanishes accordingly
[19]. Furthermore, one can easily demonstrate the symmetry
${\sigma_{\downarrow\uparrow}}(\phi)=-{\sigma_{\downarrow\uparrow}}(\pi-\phi)$,
see App. D below Eq. (41). This symmetry is readily seen in Fig. 6(c), which
shows a cut through the phase diagrams for fixed $\Delta=0$. Combined with the
symmetry
${\sigma_{\downarrow\uparrow}}(\phi)=-{\sigma_{\downarrow\uparrow}}(-\phi)$
inherited from the Haldane model, this gives the Hall drag a periodicity

$\displaystyle{\sigma_{\downarrow\uparrow}}(\phi)={\sigma_{\downarrow\uparrow}}(\phi+\pi)\
,$ (17)

apparent in Fig. 6(b). This remarkable manifestation of particle-hole
antisymmetry is in stark contrast to the pure majority case, where the Chern
number only has the trivial periodicity
$\mathcal{C}(\phi)=\mathcal{C}(\phi+2\pi)$, see Fig. 6(a).

At the special particle-hole symmetric parameter points,
$\phi=\pm\pi/2,\Delta=0$, one can also get insight into the behavior of
${\sigma_{\downarrow\uparrow}}$ to higher order in $g$ (see App. E): employing
a particle-hole transformation which also exchanges band indices of the
majority particles, it can be shown that at these points the Hall drag is
antisymmetric in $g$ to all orders. So while there is no $\mathcal{O}(g)$
contribution, and the leading order, $\mathcal{O}(g^{2})$, must vanish, at
order $\mathcal{O}(g^{3})$ the Hall drag will be nonzero.

Figure 6: Impurity Hall drag ${\sigma_{\downarrow\uparrow}}$ in the Haldane
model. (a) Majority phase diagram. $\Delta_{0}=6\sqrt{3}t^{\prime}$ is the
value of $\Delta$ where the phase transition occurs for $\phi=\pi/2$. (b)
${\sigma_{\downarrow\uparrow}}$ from numerical evaluation of Eq. (41) for
$t^{\prime}=0.2$. Cuts through the phase diagram along the dashed lines are
shown in the next panels. (c) ${\sigma_{\downarrow\uparrow}}$ as function of
$\phi$ for $\Delta=0$ and two values of $t^{\prime}$. (d)
${\sigma_{\downarrow\uparrow}}$ as function of $\Delta$ for $\phi=\pi/4$ and
same two values of $t^{\prime}$. The abscissa is rescaled by
$\Delta_{0}(t^{\prime})$.

In the numerics, the jump of ${\sigma_{\downarrow\uparrow}}$ across the
topological phase transition is again prominent, and clearly delineates the
topological phases of the parent Haldane model. Its origin is analogous to the
continuum model – it comes from a sign-changing contribution of Dirac
fermions, which becomes singular upon gap closing. The only significant
difference is that there are now two Dirac cones in the problem, but except at
the special points $\phi=0,\pi$, the gap closes at only one of them. In the
language employed for the continuum model, states near the Dirac cone with
open gap count as spectator fermions. A detailed analysis of the jump leads to
(see App. D)

$\displaystyle\Delta{\sigma_{\downarrow\uparrow}}=\Delta\mathcal{C}\cdot\frac{g^{2}n_{\downarrow}}{(2\pi)^{4}}\cdot
f(t^{\prime},\phi),$ (18)

where $f(t^{\prime},\phi)$ is a numerical function defined in Eq. (43). It
involves the remaining ${\boldsymbol{q}}$ integral, which is difficult to
evaluate analytically in the lattice case. In Fig. 7(a),
$\Delta{\sigma_{\downarrow\uparrow}}$ is depicted as a function of $\phi$. It
is maximal as $\phi\rightarrow 0^{+},\pi^{-}$, where the particle-hole
asymmetry of the dispersion (away from the Dirac points) is largest. Again,
the jump occurs on top of a smooth background contribution from the spectator
fermions, presented in Fig. 7(b). It too is maximal as $\phi\rightarrow
0^{+},\pi^{-}$, approaching $1/2\Delta{\sigma_{\downarrow\uparrow}}$: close to
these angles, the spectator contribution is almost fully determined by the
second Dirac cone, which has a very small gap. Accordingly, the values of the
sign-changing drag contribution, $\sigma_{\downarrow\uparrow,\text{Dirac}}$,
and the almost Dirac-like background contribution are the same.

Figure 7: (a) Jump of the Hall drag $\Delta{\sigma_{\downarrow\uparrow}}$ in
the Haldane model as function of $\phi$, with $t^{\prime}=0.2$ and
$\Delta=\Delta_{c}$ tuned to the transition line. The dashed line corresponds
to formula (18), points are computed numerically by evaluating Eq. (41) at two
points close to the phase boundary, with $\Delta=\Delta_{c}\pm 0.001$ (filled
circles). For comparison, a numerical evaluation with $\Delta=\Delta_{c}\pm
0.1$ is also shown (empty circles), which yields qualitative agreement only.
(b) Smooth contribution from spectator fermions, obtained numerically from
(41). Horizontal lines correspond to
$\Delta{\sigma_{\downarrow\uparrow}}(\phi=0^{+},\pi^{-})/2.$

##  V Measurement of the Hall drag

We now discuss how to detect ${\sigma_{\downarrow\uparrow}}$ experimentally.
In a solid state system, the total transversal conductivity
$\sigma_{xy,\text{tot}}$ is an easily accessible quantity, typically obtained
from a resistivity measurement. Since the majority particles form a Chern
insulator, their contribution to $\sigma_{xy,\text{tot}}$ is quantized, and
the Hall drag contribution ${\sigma_{\downarrow\uparrow}}$ can in principle be
read off by subtracting this quantized value from $\sigma_{xy,\text{tot}}$. In
practice, however, it may be necessary to use the specific parameter
dependence of ${\sigma_{\downarrow\uparrow}}$ to separate it from
$\sigma_{xy,\text{tot}}$. ${\sigma_{\downarrow\uparrow}}$ can for example be
obtained as the contribution to $\sigma_{xy,\text{tot}}$ proportional to the
impurity density $n_{\downarrow}$, or by subtracting measurements of
$\sigma_{xy,\text{tot}}$ at two particle-hole inverted points of the phase
diagram.

Chern insulators have also been successfully realized in ultracold gas
systems. Here, an established technique for measuring topological quantum
numbers [39, 40] is the in-situ observation of the center of mass displacement
of the atomic cloud upon the action of an external force. In the present
polaron context, this measurement would have to be performed in a state-
dependent manner to extract the Hall drag. In addition, one could conduct
either a state-dependent time-of-flight measurement [41, 42], or Raman
spectroscopy (as recently implemented for polarons [43]), to infer the in-trap
momentum distribution of the impurity, in view of evaluating the current
response of the impurity to an applied force.

All these transport experiments would extract the Hall drag from the linear
current response to an external, linearly polarized electric field, which is
the standard point of view. However, recent theoretical works have shown [25,
26, 27, 44, 45] that topological invariants can also be obtained from a
measurement of excitation rates to second order in the amplitudes of
circularly polarized fields, which was verified in the experiment of Ref.
[28]. For the Hall drag ${\sigma_{\downarrow\uparrow}}$, a relation to an
impurity excitation rate can be established as well, as we now show. Measuring
such excitation rates may be a simpler route to detect
${\sigma_{\downarrow\uparrow}}$ experimentally, in both ultracold gas and
solid state systems.

To set the stage, we first rephrase the results of Ref. [25] for the majority
sector (non-interacting Chern insulator). The particles are coupled to
external left or right circular polarized electrical fields:

$\displaystyle\textbf{E}_{\pm}(t)=2E\left(\cos(\omega t),\ \pm\sin(\omega
t)\right)^{T}\ ,$ (19)

with $\omega$ a fixed drive frequency. In the temporal gauge, the time-
dependent light-matter Hamiltonian reads

$\displaystyle
H_{\uparrow,\pm}(t)=\frac{2E}{\omega}\left(\hat{J}^{x}_{\uparrow}\sin(\omega
t)\mp\hat{J}^{y}_{\uparrow}\cos(\omega t)\right)\ .$ (20)

When this perturbation is switched on, particles are excited from the lower to
the upper band. One can define the associated depletion rates of initially
occupied states with momentum ${\boldsymbol{k}}$,
$\Gamma_{\uparrow,\pm}({\boldsymbol{k}},\omega)$, which depend on the
polarization of the driving field (“circular dichroism”). In Ref. [25], these
rates are obtained from Fermi’s Golden Rule. Let
$\Delta\Gamma_{\uparrow}(\omega)$ be the difference in total depletion rates
for a fixed frequency $\omega$, $\Delta\Gamma_{\uparrow}(\omega)\equiv
1/2\sum_{\boldsymbol{k}}(\Gamma_{\uparrow,+}({\boldsymbol{k}},\omega)-\Gamma_{\uparrow,-}({\boldsymbol{k}},\omega))$.
Then the Chern number $\mathcal{C}$ follows the simple relation 444Note that
we use a different sign convention for $\mathcal{C}$ than Ref. [25]:

$\displaystyle A_{0}E^{2}\mathcal{C}=-\int_{0}^{\infty}\
d\omega\Delta\Gamma_{\uparrow}(\omega)\ .$ (21)

This integration has to be understood as an average of
$\Delta\Gamma_{\uparrow}(\omega)$ over different drive frequencies, obtained
by repeating the experiment many times [28].

Figure 8: On-shell self-energy diagram. Incoming and outgoing fermion lines
represent particles from the lower band, the intermediate line a particle from
the upper band, and the wiggly lines the circularly polarized electrical
fields. The Feynman rules are explained in App. F.

For our purposes here, it is useful to rederive the result (21) from
diagrammatic perturbation theory. This is achieved by relating the depletion
rate to the on-shell retarded self-energy as

$\displaystyle\Gamma_{\pm,\uparrow}({\boldsymbol{k}},\omega)=-2\text{Im}\left[\Sigma_{\pm}(\epsilon_{\uparrow,1}({\boldsymbol{k}}),{\boldsymbol{k}};\omega)\right]\
.$ (22)

In turn, the self-energy to second order in $H_{\uparrow,\pm}$ can be
represented by the Feynman diagram of Fig. 8, plus the diagram with the
$\hat{J}^{x}_{\uparrow},\hat{J}_{\uparrow}^{y}$ vertices interchanged. The
necessary Feynman rules in energy-momentum space are easily derived from
$H_{\uparrow,\pm}$, and are detailed in App. F. There are also processes
involving $(\hat{J}^{x}_{\uparrow})^{2},(\hat{J}^{y}_{\uparrow})^{2}$, but
they cancel in $\Delta\Gamma_{\uparrow}(\omega)$. Working directly in the real
frequency space for convenience, $\Delta\Gamma_{\uparrow}(\omega)$ can then be
directly written down as

$\displaystyle\Delta\Gamma_{\uparrow}(\omega)=-\sum_{\boldsymbol{k}}\text{Im}\left[\Sigma_{+}(\epsilon_{1}({\boldsymbol{k}}),{\boldsymbol{k}};\omega)-\Sigma_{-}(\epsilon_{1}({\boldsymbol{k}}),{\boldsymbol{k}};\omega)\right]=$
$\displaystyle-\sum_{\boldsymbol{k}}\frac{E^{2}}{\omega^{2}}\text{Im}\bigg{[}\left(2iJ_{\uparrow,21}^{x}({\boldsymbol{k}})J^{y}_{\uparrow,12}({\boldsymbol{k}})-2iJ_{\uparrow,21}^{y}({\boldsymbol{k}})J_{\uparrow,12}^{x}({\boldsymbol{k}})\right)$
$\displaystyle\qquad\qquad\qquad\times\frac{1}{\omega+\epsilon_{\uparrow,1}({\boldsymbol{k}})-\epsilon_{\uparrow,2}({\boldsymbol{k}})+i0^{+}}\bigg{]}\
.$ (23)

Integrating over $\omega$, we find:

$\displaystyle\int_{0}^{\infty}d\omega\Delta\Gamma_{\uparrow}(\omega)=$ (24)
$\displaystyle\frac{4\pi E^{2}A_{0}}{(2\pi)^{2}}\int
d{\boldsymbol{k}}\int_{0}^{\infty}d\omega\frac{\delta(\omega-(\epsilon_{\uparrow,2}({\boldsymbol{k}})-\epsilon_{\uparrow,1}({\boldsymbol{k}})))}{\omega^{2}}$
$\displaystyle\times\text{Im}\left[J_{\uparrow,12}^{x}({\boldsymbol{k}})J_{\uparrow,21}^{y}({\boldsymbol{k}})\right]\overset{\eqref{sigmaCrel}}{=}-A_{0}E^{2}\mathcal{C}\
,$

in agreement with Eq. (21).

To summarize, we have related the majority Chern number to the differential
depletion rate of filled states from the lower band when the system is
subjected to a circular perturbation. We can now extend this idea to the
impurity case. We consider our previous interacting majority-impurity setup,
with a small number of impurities prepared in the lower band, and couple both
majority and impurity particles to the circular fields. On their own, the
impurities would not experience a differential depletion because of the time
reversal invariance of the impurity Hamiltonian. Only due the interaction with
the majority particles such differential depletion will set in, corresponding
to occupation of higher momentum states. Note that, for strong impurity-
majority interactions, it will rather be polaronic (dressed impurity) states
which are depleted. For weak coupling, however, such band-dressing effects can
be neglected (to order $\mathcal{O}(g^{2})$), and we can think in terms of
bare impurities in lieu of polarons. In technical terms, our Feynman diagrams
will not contain any impurity self-energy insertions.

Let us couple the impurities to the circular fields in the same way as the
majority particles, Eq. (20). We consider the depletion rate of the filled
impurity state with vanishing momentum
$\Gamma_{\downarrow,\pm}({\boldsymbol{{0}}},\omega)\equiv\Gamma_{\downarrow,\pm}(\omega)$,
which is of most interest when the impurity density is small. Since non-
vanishing contributions to $\Delta\Gamma_{\downarrow}(\omega)$ must involve
majority scattering, to order $\mathcal{O}(g^{2})$ there are two classes of
relevant diagrams; representative diagrams are shown in Fig. 9.

Figure 9: Non-vanishing contributions to the impurity depletion rate
$\Gamma_{\downarrow,\pm}({\boldsymbol{{0}}},\omega)$. Panels (a), (b):
Diagrams not related to the drag, which are particle-hole symmetric. Panels
(c), (d): Diagrams related to the drag. These two diagrams differ in the
orientation of the field lines and the band index structure of the majority
particles.

Consider first the two diagrams 9(a), 9(b) in the top row of the Figure. These
diagrams describe processes where only the majority particles are excited by
the external fields. Since they do not involve an impurity current, they are
not related to the drag. Two additional diagrams where the direction of the
external field lines is inverted can be drawn as well.

The structural difference between Fig. 9(a) and 9(b) is the orientation of the
majority lines, which maps to an 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 up rather than cancel, since they do not contain an
impurity current operator, $J_{\downarrow}({\boldsymbol{q}})$, which is odd in
${\boldsymbol{q}}$. Therefore, as can be verified by a straightforward
evaluation (cf. App. F, Eq. (52)), the total contribution
$\Delta\Gamma_{\downarrow,\text{ph}}$ of these diagrams obeys
$\Delta\Gamma_{\downarrow,\text{ph}}(\phi)=\Delta\Gamma_{\downarrow,\text{ph}}(\pi-\phi)$
for the Haldane and
$\Delta\Gamma_{\downarrow,\text{ph}}(d_{2})=\Delta\Gamma_{\downarrow,\text{ph}}(-d_{2})$
for the continuum model. As a result, in an experiment these processes can be
projected out by subtracting
$\Delta\Gamma_{\downarrow,\text{ph}}(\phi)-\Delta\Gamma_{\downarrow,\text{ph}}(\pi-\phi)$,
which leaves out only the antisymmetric drag contribution. Another way to
separate $\Delta\Gamma_{\downarrow,\text{ph}}$ from the drag is to have a
different coupling constant between external field and impurities, which is
feasible in the ultracold gas setup where the circular perturbation can for
example be implemented by lattice shaking [47, 28]. Since
$\Delta\Gamma_{\downarrow,\text{ph}}$ is independent of the coupling to the
impurities, it can again be eliminated by subtracting measurements obtained
for two different impurity couplings.

Let us assume either such elimination implicitly, and move on to the two
diagrams of Fig. 9(c), 9(d). In essence, they correspond to the drag
transconductivity diagram of Fig. 3 (top), with the central (red) impurity
line cut. The two other diagrams in this class have crossed interaction lines,
akin to the “crossed” diagrams of Fig. 3 (bottom). Evaluation of these four
diagrams is straightforward, see App. F. Summation over the filled impurity
states simply yields:

$\displaystyle\sum_{{\boldsymbol{p}},\text{filled}}\Gamma_{\downarrow,\pm}({\boldsymbol{p}},\omega)\simeq\sum_{{\boldsymbol{p}},\text{filled}}\Gamma_{\downarrow,\pm}(\omega)=A_{0}n_{\downarrow}\Gamma_{\downarrow,\pm}(\omega)\
.$ (25)

For the integrated differential depletion rate, one then finds

$\displaystyle\int_{0}^{\infty}d\omega\Delta\Gamma_{\downarrow,xy}(\omega)=2\pi
A_{0}E^{2}{\sigma_{\downarrow\uparrow}}\ ,$ (26)

as naively expected from Eq. (24). However, the impurity depletion rate also
receives contribution from processes involving the currents
$\hat{J}_{\downarrow}^{y}$, $\hat{J}_{\uparrow}^{x}$. 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 differential rate, since
$\sigma_{xy,\downarrow\uparrow}=-\sigma_{yx,\downarrow\uparrow}$ for both the
continuum and the Haldane model, as one can check easily. Modulo the
antisymmetrization discussed above, we therefore have

$\displaystyle{\sigma_{\downarrow\uparrow}}=\frac{1}{4\pi
A_{0}E^{2}}\int_{0}^{\infty}d\omega\Delta\Gamma_{\downarrow}(\omega)\ .$ (27)

This result can also be rephrased in terms of excitation instead of depletion
rates. Since the impurities are initially prepared at the bottom of the band,
one can write

$\displaystyle\int_{0}^{\infty}d\omega\Delta\Gamma_{\downarrow}(\omega)=\sum_{{\boldsymbol{q}}>0}\
\int_{0}^{\infty}d\omega\Delta\Gamma_{\downarrow,\text{exc}}({\boldsymbol{q}},\omega)\
,$ (28)

meaning that the impurities are excited into states with higher momentum which
are initially empty. These ${\boldsymbol{q}}$-states correspond to the
intermediate impurity lines in Fig. 9. Via Eq. (27) we can then define a
${\boldsymbol{q}}$-resolved impurity drag as

$\displaystyle{\sigma_{\downarrow\uparrow}}\equiv\sum_{{\boldsymbol{q}}>0}{\sigma_{\downarrow\uparrow}}({\boldsymbol{q}})\
.$ (29)

This provides an alternative view on, say, the topological jump
$\Delta{\sigma_{\downarrow\uparrow}}$. For the Haldane model, it can be
phrased as $\Delta{\sigma_{\downarrow\uparrow}}=\Delta\mathcal{C}\int
d{\boldsymbol{q}}f_{\text{jump}}({\boldsymbol{q}})$, where
$f_{\text{jump}}({\boldsymbol{q}})$ is a known function, see Eqs. (18), (43).
If the excitation rates defined in Eq. (28) can be experimentally detected in
${\boldsymbol{q}}$-resolved fashion (for instance with band mapping techniques
[48, 49, 50]), so can the ${\boldsymbol{q}}$-resolved impurity drag
${\sigma_{\downarrow\uparrow}}({\boldsymbol{q}})$. Measuring
${\sigma_{\downarrow\uparrow}}({\boldsymbol{q}})$ at two points in the phase
diagram close to the topological boundary then gives direct access to
$f_{\text{jump}}({\boldsymbol{q}})$. Taken the other way around, supposing
that $f_{\text{jump}}({\boldsymbol{q}})$ is known for the model realized in
the experiment, at each ${\boldsymbol{q}}$-point an independent estimate of
the change in Chern number across the phase transition $\Delta\mathcal{C}$ is
possible.

##  VI Conclusions

In this work we have studied to which extent a topologically trivial impurity
can be Hall-dragged by majority excitations in a Chern insulator, looking at
two different models in a controlled perturbative setting. Since the impurity
Hall drag is sensitive to the dispersion of the majority particles and holes,
there is no one-to-one correspondence to the Chern number; nevertheless, the
change in Chern number across a topological transition is clearly reflected by
a discontinuous jump in the drag transconductivity
${\sigma_{\downarrow\uparrow}}$. This jump arises from the integrated singular
Berry curvature of the majority fermions.The transconductivity can be
extracted either from transport experiments, or from a measurement of impurity
excitation rates upon driving the system by a circularly polarized field.

A worthwhile goal for future study is the extension to the strong-coupling
limit, in particular the analysis of impurity-majority bound state formation.
These bound states may have rather rich physics: They could inherit the
topological characteristics of the majority particles [14, 15], have opposite
chirality as found for Haldane model in the two-body limit [51], or even be
topological when the single-particle state are trivial [52, 53, 54].

## Acknowledgments

We thank A. Kamenev for helpful discussions. D.P. acknowledges funding by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy – EXC-2111 – 390814868, and is particularly
grateful to the Max-Planck-Institute for the Physics of Complex Systems
Dresden (MPIPKS) for hospitality during the intermediate stage of this
project. 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 Independent Research Fund Denmark-Natural Sciences via Grant No.
DFF-8021-00233B, and US Army CCDC Atlantic Basic and Applied Research via
grant W911NF-19-1-0403. M.G. has been supported by the Israel Science
Foundation (Grant No. 227/15) and the US-Israel Binational Science Foundation
(Grant No. 2016224).

##  Appendix A Basis rotation

To evaluate the conductivities, it is convenient to work in the diagonal band
basis, introducing a diagonalizing unitary matrix
$U_{\uparrow}({\boldsymbol{k}})$

$\displaystyle
U_{\uparrow}^{\dagger}({\boldsymbol{k}})H_{\uparrow}({\boldsymbol{k}})U_{\uparrow}({\boldsymbol{k}})=\text{diag}(\epsilon_{\uparrow,1}({\boldsymbol{k}}),\epsilon_{\uparrow,2}({\boldsymbol{k}}))$
(30) $\displaystyle
U_{\uparrow}({\boldsymbol{k}})=\begin{pmatrix}U_{\uparrow,A1}({\boldsymbol{k}})&U_{\uparrow,A2}({\boldsymbol{k}})\\\
U_{\uparrow,B1}({\boldsymbol{k}})&U_{\uparrow,B2}({\boldsymbol{k}})\end{pmatrix},$
(31) $\displaystyle
U_{\uparrow,A1}({\boldsymbol{k}})=\frac{h_{3}({\boldsymbol{k}})-h({\boldsymbol{k}})}{\sqrt{2h({\boldsymbol{k}})(h({\boldsymbol{k}})-h_{3}({\boldsymbol{k}}))}}$
$\displaystyle
U_{\uparrow,A2}({\boldsymbol{k}})=\frac{h_{3}({\boldsymbol{k}})+h({\boldsymbol{k}})}{\sqrt{2h({\boldsymbol{k}})(h({\boldsymbol{k}})+h_{3}({\boldsymbol{k}}))}}$
$\displaystyle
U_{\uparrow,B1}({\boldsymbol{k}})=\frac{h_{1}({\boldsymbol{k}})+ih_{2}({\boldsymbol{k}})}{\sqrt{2h({\boldsymbol{k}})(h({\boldsymbol{k}})-h_{3}({\boldsymbol{k}}))}},$
$\displaystyle
U_{\uparrow,B2}({\boldsymbol{k}})=\frac{h_{1}({\boldsymbol{k}})+ih_{2}({\boldsymbol{k}})}{\sqrt{2h({\boldsymbol{k}})(h({\boldsymbol{k}})+h_{3}({\boldsymbol{k}}))}}\
,$

where $A,B$ refer to the sublattice- and $1,2$ to the diagonal band basis. The
same expressions apply for the Haldane model as well.

In the band basis, the second-quantized current operator is given by

$\displaystyle\hat{J}^{x/y}_{\uparrow}$
$\displaystyle=\sum_{\boldsymbol{k}}c^{\dagger}_{\uparrow,\alpha}({\boldsymbol{k}})J^{x/y}_{\uparrow,\alpha\beta}({\boldsymbol{k}})c_{\uparrow,\beta}({\boldsymbol{k}}),$
(32)

with matrix elements

$\displaystyle J_{\uparrow}^{x}({\boldsymbol{k}})$
$\displaystyle=U_{\uparrow}^{\dagger}({\boldsymbol{k}})J_{\uparrow}^{x,0}({\boldsymbol{k}})U_{\uparrow}({\boldsymbol{k}})$
(33) $\displaystyle J_{\uparrow}^{x,0}({\boldsymbol{k}})$
$\displaystyle=\frac{\partial H_{\uparrow}({\boldsymbol{k}})}{\partial
k_{x}}=\sigma_{x}+2k_{x}(d_{1}\sigma_{z}+d_{2}\mathbbm{1}),$

and likewise for $J_{\uparrow}^{y}({\boldsymbol{k}})$.

##  Appendix B Evaluation of the drag diagrams in the continuum model

Let us start by considering the first “direct” diagram in Fig. 3 with majority
band indices $\alpha=1,\beta=2$. Its contribution to the Matsubara correlator
$-\braket{\hat{J}_{\downarrow}^{x}\hat{J}_{\uparrow}^{y}}(i\Omega)$, to be
denoted by $\mathcal{P}_{1}(i\Omega)$, reads

$\displaystyle\mathcal{P}_{1}(i\Omega)=-g^{2}n_{\downarrow}A_{0}\int_{k,q}\int\frac{d\tilde{\omega}}{2\pi}J_{\uparrow,12}^{y}({\boldsymbol{k}})W^{22}({\boldsymbol{k}}-{\boldsymbol{q}},{\boldsymbol{q}})W^{21}({\boldsymbol{k}},-{\boldsymbol{q}})J^{x}_{\downarrow}({\boldsymbol{q}})$
(34)
$\displaystyle\frac{1}{i(\Omega+\omega_{q})-\epsilon_{\downarrow}({\boldsymbol{q}})}\frac{1}{i\omega_{q}-\epsilon_{\downarrow}({\boldsymbol{q}})}\frac{1}{i(\omega_{q}+\tilde{\omega})+0^{+}}\frac{1}{i(\Omega+\omega_{k})-{\epsilon_{\uparrow,1}}({\boldsymbol{k}})}\frac{1}{i\omega_{k}-{\epsilon_{\uparrow,2}}({\boldsymbol{k}})}\frac{1}{i(\omega_{k}+\tilde{\omega})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}}-{\boldsymbol{q}})}\
,$

where $0^{+}$ in the third impurity propagator ensures the correspondence to
filled states. Evaluating the frequency integrals we find:

$\displaystyle\mathcal{P}_{1}(i\Omega)=-g^{2}n_{\downarrow}A_{0}\int\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}\frac{d{\boldsymbol{q}}}{(2\pi)^{2}}J_{\uparrow,12}^{y}({\boldsymbol{k}})W^{22}({\boldsymbol{k}}-{\boldsymbol{q}},{\boldsymbol{q}})W^{21}({\boldsymbol{k}},-{\boldsymbol{q}})J^{x}_{\downarrow}({\boldsymbol{q}})$
(35)
$\displaystyle\frac{1}{{\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}}-{\boldsymbol{q}})-\epsilon_{\downarrow}({\boldsymbol{q}})}\frac{1}{-i\Omega+{\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}}-{\boldsymbol{q}})-\epsilon_{\downarrow}({\boldsymbol{q}})}\frac{1}{-i\Omega+{\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}})}\
.$

Upon analytical continuation, $i\Omega\rightarrow\omega$, only the
$\mathcal{O}(\omega)$ part contributes to the static drag as in the non-
interacting case. With Eq. (9), we get:

$\displaystyle\sigma_{\downarrow\uparrow,1}=ig^{2}n_{\downarrow}\int\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}\frac{d{\boldsymbol{q}}}{(2\pi)^{2}}J^{y}_{\uparrow,12}({\boldsymbol{k}})W^{22}({\boldsymbol{k}}-{\boldsymbol{q}},{\boldsymbol{q}})W^{21}({\boldsymbol{k}},-{\boldsymbol{q}})\frac{q_{x}}{M}\frac{1}{\left({\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}})\right)^{2}}d({\boldsymbol{k}},{\boldsymbol{q}}),$
(36)

with $d({\boldsymbol{k}},{\boldsymbol{q}})$ as defined in Eq. (11). The
remaining three contributions to $\sigma_{\downarrow\uparrow}$ have the
following structure: The direct diagram with majority indices
$\alpha=2,\beta=1$ leads to Eq. (36) with $A\equiv
J^{y}_{\uparrow,12}({\boldsymbol{k}})W^{22}({\boldsymbol{k}}-{\boldsymbol{q}},{\boldsymbol{q}})W^{21}({\boldsymbol{k}},-{\boldsymbol{q}})$
replaced by
$B\equiv-J^{y}_{\uparrow,21}({\boldsymbol{k}})W^{12}({\boldsymbol{k}}-{\boldsymbol{q}},{\boldsymbol{q}})W^{22}({\boldsymbol{k}},-{\boldsymbol{q}})$;
using elementary properties of unitary matrices, one can show that
$B=-\overline{A}$ (with $\overline{A}$ the complex conjugate of $A$), thus
yielding the part $\propto d({\boldsymbol{k}},{\boldsymbol{q}})$ of Eq. (10)
in the main text. The remaining “crossed” diagram of Fig. 3 likewise generates
the part $\propto c({\boldsymbol{k}},{\boldsymbol{q}})$.

##  Appendix C Jump of the Hall drag in the continuum model

To derive the jump from Eq. (10), we need to project on the part of the
${\boldsymbol{k}}$-integrand corresponding to the Dirac fermions, which
becomes singular at ${\boldsymbol{k}}=0$ as $m\rightarrow 0$. This can be done
by setting ${\boldsymbol{k}}=0$ in all regular parts. The last factor in the
integrand becomes

$\displaystyle
d({\boldsymbol{k}},{\boldsymbol{q}})+c({\boldsymbol{k}},{\boldsymbol{q}})\rightarrow$
(37)
$\displaystyle\left(\frac{1}{(\epsilon_{\downarrow}({\boldsymbol{q}})+{\epsilon_{\uparrow,2}}({\boldsymbol{q}}))^{2}}-\frac{1}{(\epsilon_{\downarrow}({\boldsymbol{q}})-{\epsilon_{\uparrow,1}}({\boldsymbol{q}}))^{2}}\right)\
.$

In the part involving interaction matrices $W$, it is useful to rewrite

$\displaystyle
W^{22}({\boldsymbol{k}}-{\boldsymbol{q}},{\boldsymbol{q}})W^{21}({\boldsymbol{k}},-{\boldsymbol{q}})\overset{\eqref{Hint}}{=}$
(38) $\displaystyle
U^{\dagger}_{\uparrow,2n}({\boldsymbol{k}})U_{\uparrow,n2}({\boldsymbol{k}}-{\boldsymbol{q}})U^{\dagger}_{\uparrow,2m}({\boldsymbol{k}}-{\boldsymbol{q}})U_{\uparrow,m1}({\boldsymbol{k}})\rightarrow$
$\displaystyle
U^{\dagger}_{\uparrow,2n}({\boldsymbol{k}})U_{\uparrow,n2}(-{\boldsymbol{q}})U^{\dagger}_{\uparrow,2m}(-{\boldsymbol{q}})U_{\uparrow,m1}({\boldsymbol{k}})=$
$\displaystyle\left(U^{\dagger}_{\uparrow}({\boldsymbol{k}})V({\boldsymbol{q}})U_{\uparrow}({\boldsymbol{k}})\right)_{21},$
$\displaystyle V({\boldsymbol{q}})_{nm}\equiv
U_{\uparrow,n2}(-{\boldsymbol{q}})U^{\dagger}_{\uparrow,2m}(-{\boldsymbol{q}})\
,$

where $n,m$ are sublattice indices, and in the second step we have only kept
the singular ${\boldsymbol{k}}$ dependence. $V({\boldsymbol{q}})$ is a
hermitian matrix, and so can be expanded as a linear combination of the unit
and Pauli matrices with real coefficients. Then it is easy to show that only
the contribution $\propto\sigma_{x}$ survives the integration in Eq. (10),
while the rest either does not contribute to the required imaginary part or is
antisymmetric in $k_{x}$. Therefore, we can write

$\displaystyle\left(U^{\dagger}_{\uparrow}({\boldsymbol{k}})V({\boldsymbol{q}})U_{\uparrow}({\boldsymbol{k}})\right)_{21}\hat{=}$
(39)
$\displaystyle\left(U^{\dagger}_{\uparrow}({\boldsymbol{k}})\sigma_{x}U_{\uparrow}({\boldsymbol{k}})\right)_{21}\text{Re}\left[V({\boldsymbol{q}})_{12}\right]=$
$\displaystyle\left(U^{\dagger}_{\uparrow}({\boldsymbol{k}})J^{x,0}_{\uparrow,\text{Dirac}}U_{\uparrow}({\boldsymbol{k}})\right)_{21}\frac{-q_{x}}{2q\sqrt{1+(d_{1}q)^{2}}}\
,$

where in the last step we identified the current operator of the Dirac
fermions in the sublattice basis, $\sigma_{x}=J^{x,0}_{\uparrow,\text{Dirac}}$
(cf. Eq. (1)), and wrote out $V({\boldsymbol{q}})$ by inserting matrix
elements of $U_{\uparrow}(-{\boldsymbol{q}})$ from App. A. Inserting Eqs.
(37), (39) into Eq. (10), we can write the sign-changing Dirac part of the
Hall drag as shown in Eq. (12) of the main text.

##  Appendix D Impurity Hall drag and jump in the Haldane model

In the Haldane model, the on-site interaction is defined by (cf. Eq. (8))

$\displaystyle
H_{\text{int}}=\frac{g}{A_{0}}\sum_{\ell=A,B}\sum_{{\boldsymbol{k}},{\boldsymbol{p}},{\boldsymbol{q}}}c^{\dagger}_{\uparrow,\ell}({\boldsymbol{k}}+{\boldsymbol{q}})c_{\uparrow,\ell}({\boldsymbol{k}})c^{\dagger}_{\downarrow,\ell}({\boldsymbol{p}}-{\boldsymbol{q}})c_{\downarrow,\ell}({\boldsymbol{p}})=\frac{g}{A_{0}}\sum_{{\boldsymbol{k}},{\boldsymbol{p}},{\boldsymbol{q}}}c^{\dagger}_{\uparrow,\alpha}({\boldsymbol{k}}+{\boldsymbol{q}})c_{\uparrow,\beta}({\boldsymbol{k}})c^{\dagger}_{\downarrow,1}({\boldsymbol{p}}-{\boldsymbol{q}})c_{\downarrow,1}({\boldsymbol{p}})W_{\alpha\beta}({\boldsymbol{k}},{\boldsymbol{p}},{\boldsymbol{q}})\
,$ $\displaystyle
W_{\alpha\beta}({\boldsymbol{k}},{\boldsymbol{p}},{\boldsymbol{q}})=\sum_{\ell=A,B}\overline{U}_{\uparrow,\ell\alpha}({\boldsymbol{k}}+{\boldsymbol{q}})U_{\uparrow,\ell\beta}({\boldsymbol{k}})\overline{U}_{\downarrow,\ell
1}({\boldsymbol{p}}-{\boldsymbol{q}})U_{\downarrow,\ell 1}({\boldsymbol{p}})\
.$ (40)

In Eq. (40), 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, and results in

$\displaystyle\sigma_{\downarrow\uparrow}=$ (41)
$\displaystyle-2g^{2}n_{\downarrow}\int\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}\frac{d{\boldsymbol{q}}}{(2\pi)^{2}}\
\text{Im}\left\\{J^{y}_{\uparrow,12}({\boldsymbol{k}})W^{22}({\boldsymbol{k}}-{\boldsymbol{q}},{\boldsymbol{q}},{\boldsymbol{q}})W^{21}({\boldsymbol{k}},{\boldsymbol{{0}}},-{\boldsymbol{q}})\right\\}J^{x}_{\downarrow,11}({\boldsymbol{q}})\frac{1}{\left({\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}})\right)^{2}}\left(d({\boldsymbol{k}},{\boldsymbol{q}})+c({\boldsymbol{k}},{\boldsymbol{q}})\right),$

with $J_{\downarrow,11}^{x}(q)$ the impurity current operator in the band
basis (taking into account lower band contributions only), and $c,d$ as in Eq.
(11), only replacing the single band energy of the continuum model
$\epsilon_{\downarrow}({\boldsymbol{q}})$ by the lower band energy
$\epsilon_{\downarrow,1}({\boldsymbol{q}})$.

From Eq. (41) one can readily derive the additional symmetry
${\sigma_{\downarrow\uparrow}}(\phi)=-{\sigma_{\downarrow\uparrow}}(\pi-\phi)$
mentioned in the main text. In the majority Hamiltonian
$H_{\uparrow}({\boldsymbol{k}})$,
$h_{0}({\boldsymbol{k}};\phi)=-h_{0}({\boldsymbol{k}};\pi-\phi)$, while the
other coefficients are invariant under such reflection. As a result, one finds
$c({\boldsymbol{k}},{\boldsymbol{q}};\phi)=-d({\boldsymbol{k}},{\boldsymbol{q}};\pi-\phi)$.
All other elements of Eq. (41) do not change, which shows the property as
claimed.

To evaluate the jump of the Hall drag $\Delta{\sigma_{\downarrow\uparrow}}$ in
the Haldane model in analogy with Sec. III, let us focus on the transition
line, $\Delta_{c}=6\sqrt{3}t^{\prime}\sin(\phi)$, where the gap closes at the
Dirac point ${\boldsymbol{k}}_{A}=(0,4\pi/3\sqrt{3})^{T}$. Since
${\sigma_{\downarrow\uparrow}}$ is symmetric in $\Delta$, for a given value of
$\phi$ the value of $\Delta{\sigma_{\downarrow\uparrow}}$ at $-\Delta_{c}$ is
the same. To extract the singular Dirac contribution at
${\boldsymbol{k}}_{A}$, we let
${\boldsymbol{k}}\rightarrow{\boldsymbol{k}}_{A}$ in all regular parts of Eq.
(41). In this limit,

$\displaystyle J_{\uparrow}^{x}({\boldsymbol{k}})\rightarrow
U_{\uparrow}^{\dagger}({\boldsymbol{k}})\frac{3}{2}\sigma_{y}U_{\uparrow}({\boldsymbol{k}})\equiv
J_{\uparrow,\text{Dirac}}^{x}({\boldsymbol{k}})$ (42)

This current can be extracted from the interaction part of Eq. (41) as in Sec.
III, which allows to write the Dirac part of the Hall drag as

$\displaystyle\sigma_{\downarrow\uparrow,\text{Dirac}}=\frac{g^{2}n_{\downarrow}}{(2\pi)^{4}}\int\frac{d{\boldsymbol{k}}}{\pi}\
\text{Im}\left\\{J_{\uparrow,\text{Dirac},12}^{y}({\boldsymbol{k}})J_{\uparrow,\text{Dirac},21}^{x}({\boldsymbol{k}})\right\\}\cdot\frac{1}{\left({\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}})\right)^{2}}\cdot
f(t^{\prime},\phi)\ ,$ (43) $\displaystyle
f(t^{\prime},\phi)\equiv-\frac{4\pi}{3}\int
d{\boldsymbol{q}}J_{\downarrow}^{x}({\boldsymbol{q}})\left(\frac{1}{\left({\epsilon_{\uparrow,1}}({\boldsymbol{k}}_{A})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}}_{A}-{\boldsymbol{q}})-{\epsilon_{\downarrow,1}}({\boldsymbol{q}})\right)^{2}}-\frac{1}{\left({\epsilon_{\uparrow,1}}({\boldsymbol{k}}_{A}-{\boldsymbol{q}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}}_{A})-{\epsilon_{\downarrow,1}}({\boldsymbol{q}})\right)^{2}}\right)\cdot$
$\displaystyle\qquad\qquad\qquad\text{Im}\left\\{U_{\uparrow,A2}({\boldsymbol{k}}_{A}-{\boldsymbol{q}})U^{\dagger}_{\downarrow,1A}({\boldsymbol{{0}}})U_{\downarrow,A1}({\boldsymbol{q}})U^{\dagger}_{\uparrow,2B}({\boldsymbol{k}}_{A}-{\boldsymbol{q}})U_{\downarrow,B1}({\boldsymbol{{0}}})U^{\dagger}_{\downarrow,1B}({\boldsymbol{q}})\right\\}\quad$

Again, the ${\boldsymbol{k}}$ and ${\boldsymbol{q}}$ integrals have
factorized, and the ${\boldsymbol{k}}$ integral gives $\pm 1/2$. This yields a
value of the jump as in Eq. (18) of the main text. The remaining
${\boldsymbol{q}}$ integral has to be evaluated numerically.

##  Appendix E Antisymmetry of the hall drag as function of $g$ in the Haldane
model with $\phi=\pm\pi/2,\Delta=0$.

Here we show that the Hall drag ${\sigma_{\downarrow\uparrow}}$ in the Haldane
model, with parameters $\phi=\pm\pi/2,\Delta=0$, is antisymmetric in the
impurity-majority coupling $g$ to all orders. We work in the diagonal band
frame, and perform a particle-hole transformation which also exchanges the
band indices:

$\displaystyle b_{\tilde{\alpha}}({\boldsymbol{k}})\equiv
c^{\dagger}_{\uparrow,\alpha}(-{\boldsymbol{k}}),\quad{b}^{\dagger}_{\tilde{\alpha}}({\boldsymbol{k}})\equiv{c}_{\uparrow,\alpha}(-{\boldsymbol{k}}),\quad{\tilde{1}}\equiv
2,\quad\tilde{2}\equiv 1\ .$ (44)

Due to particle-hole symmetry for $\phi=\pm\pi/2$, the form of the non-
interacting majority Hamiltonian is invariant under this transformation (up to
a constant):

$\displaystyle H_{\uparrow}$
$\displaystyle=\sum_{\boldsymbol{k}}c^{\dagger}_{\uparrow,\alpha}({\boldsymbol{k}})\epsilon_{\alpha}({\boldsymbol{k}})c_{\uparrow,\alpha}({\boldsymbol{k}})$
(45)
$\displaystyle=\sum_{\boldsymbol{k}}b^{\dagger}_{\tilde{\alpha}}(-{\boldsymbol{k}})\left[-\epsilon_{\alpha}({\boldsymbol{k}})\right]b_{\tilde{\alpha}}(-{\boldsymbol{k}})+\text{const.}$
$\displaystyle=\sum_{\boldsymbol{k}}b^{\dagger}_{\alpha}({\boldsymbol{k}})\epsilon_{\alpha}({\boldsymbol{k}})b_{\alpha}({\boldsymbol{k}})+\text{const.}\
,$

where
$\epsilon_{\alpha}({\boldsymbol{k}})=-\epsilon_{\tilde{\alpha}}(-{\boldsymbol{k}})$
was used. However, the interaction term acquires a minus sign under the
variable transformation (44):

$\displaystyle H_{\text{int}}$
$\displaystyle=\frac{g}{A_{0}}\sum_{{\boldsymbol{k}},{\boldsymbol{p}},{\boldsymbol{q}}}c^{\dagger}_{\uparrow,\alpha}({\boldsymbol{k}}+{\boldsymbol{q}})c_{\uparrow,\beta}({\boldsymbol{k}})c^{\dagger}_{\downarrow,1}({\boldsymbol{p}}-{\boldsymbol{q}})c_{\downarrow,1}({\boldsymbol{p}})W_{\alpha\beta}({\boldsymbol{k}},{\boldsymbol{p}},{\boldsymbol{q}})$
(46)
$\displaystyle=-\frac{g}{A_{0}}\sum_{{\boldsymbol{k}},{\boldsymbol{p}},{\boldsymbol{q}}}b^{\dagger}_{\tilde{\beta}}(-{\boldsymbol{k}}){b}_{\tilde{\alpha}}(-{\boldsymbol{k}}-{\boldsymbol{q}})c^{\dagger}_{\downarrow,1}({\boldsymbol{p}}-{\boldsymbol{q}}){c}_{\downarrow,1}({\boldsymbol{p}})W_{\alpha\beta}({\boldsymbol{k}},{\boldsymbol{p}},{\boldsymbol{q}})+\text{const.}$
$\displaystyle=-\frac{g}{A_{0}}\sum_{{\boldsymbol{k}},{\boldsymbol{p}},{\boldsymbol{q}}}{b}^{\dagger}_{\alpha}({\boldsymbol{k}}+{\boldsymbol{q}}){b}_{\beta}({\boldsymbol{k}})c^{\dagger}_{\downarrow,1}({\boldsymbol{p}}-{\boldsymbol{q}}){c}_{\downarrow,1}({\boldsymbol{p}})W_{\tilde{\beta}\tilde{\alpha}}(-{\boldsymbol{k}}-{\boldsymbol{q}},{\boldsymbol{p}},{\boldsymbol{q}})+\text{const.}$
$\displaystyle=-\frac{g}{A_{0}}\sum_{{\boldsymbol{k}},{\boldsymbol{p}},{\boldsymbol{q}}}b^{\dagger}_{\alpha}({\boldsymbol{k}}+{\boldsymbol{q}})b_{\beta}({\boldsymbol{k}})c^{\dagger}_{\downarrow,1}({\boldsymbol{p}}-{\boldsymbol{q}})c_{\downarrow,1}({\boldsymbol{p}})W_{\alpha\beta}({\boldsymbol{k}},{\boldsymbol{p}},{\boldsymbol{q}})+\text{const.}\
.$

The unimportant additional terms are constant in the majority sector. In the
last step, we used
$W_{\tilde{\beta}\tilde{\alpha}}(-{\boldsymbol{k}}-{\boldsymbol{q}},{\boldsymbol{p}},{\boldsymbol{q}})=W_{\alpha\beta}({\boldsymbol{k}},{\boldsymbol{p}},{\boldsymbol{q}})$.
This can be easily shown by inserting the matrix elements from Eqs. (31),
(16), but requires $h_{3}({\boldsymbol{k}})=-h_{3}(-{\boldsymbol{k}})$, which
is only fulfilled for $\Delta=0$ (and is violated in the continuum model).

Last, the required majority current operator transforms as

$\displaystyle
J_{\uparrow,\alpha\beta}^{y}({\boldsymbol{k}})=\sum_{\boldsymbol{k}}c^{\dagger}_{\uparrow,\alpha}({\boldsymbol{k}})U^{\dagger}_{\uparrow,\alpha
n}({\boldsymbol{k}})\left[J_{y,\uparrow}^{0}({\boldsymbol{k}})\right]_{nm}U_{\uparrow,m\beta}({\boldsymbol{k}})c_{\uparrow,\beta}({\boldsymbol{k}}),\quad\left[J_{y,\uparrow}^{0}({\boldsymbol{k}})\right]_{nm}=\left[\partial_{k_{y}}H_{\uparrow}({\boldsymbol{k}})\right]_{nm}\
,$ (47) $\displaystyle
J_{\uparrow,\alpha\beta}^{y}({\boldsymbol{k}})=-\sum_{{\boldsymbol{k}}}{b}^{\dagger}_{\tilde{\beta}}(-{\boldsymbol{k}})U^{\dagger}_{\uparrow,\alpha
n}({\boldsymbol{k}})\left[J_{y,\uparrow}^{0}({\boldsymbol{k}})\right]_{nm}U_{\uparrow,m\beta}({\boldsymbol{k}})b_{\tilde{\alpha}}(-{\boldsymbol{k}})+\text{const.}$
$\displaystyle\quad\quad\quad\ \
=-\sum_{{\boldsymbol{k}}}{b}^{\dagger}_{\alpha}({\boldsymbol{k}})U^{T}_{\uparrow,\tilde{\alpha}m}(-{\boldsymbol{k}})\left[J_{y,\uparrow}^{0}(-{\boldsymbol{k}})\right]^{T}_{mn}\overline{U}_{\uparrow,n\tilde{\beta}}(-{\boldsymbol{k}})b_{\beta}({\boldsymbol{k}})+\text{const.}\
.$

Again, inserting matrix elements one can show that

$\displaystyle
U^{T}_{\uparrow,\tilde{\alpha}m}(-{\boldsymbol{k}})\left[J_{y,\uparrow}^{0}(-{\boldsymbol{k}})\right]^{T}_{mn}\overline{U}_{\uparrow,n\tilde{\beta}}(-{\boldsymbol{k}})=$
$\displaystyle U_{\uparrow,\alpha
m}^{\dagger}({\boldsymbol{k}})\left[J_{y,\uparrow}^{0}({\boldsymbol{k}})\right]_{mn}U_{\uparrow,n\beta}({\boldsymbol{k}}),$

and the majority current changes sign. In conclusion, for
$\phi=\pm\pi/2,\Delta=0$ this proves the antisymmetry

$\displaystyle{\sigma_{\downarrow\uparrow}}(g)=-{\sigma_{\downarrow\uparrow}}(-g)\
,$ (48)

as claimed in the main text.

##  Appendix F ${\sigma_{\downarrow\uparrow}}$ from circular dichroism:
Technical details

The Feynman rules for the perturbation $H_{\uparrow,\pm}(t)$ of Eq. (20) in
the energy-momentum domain are easily derived from Wick’s theorem. They read:

  * •

Each current vertex comes with a factor $E/\omega$.

  * •

If an incoming (outgoing) electrical field line couples to a $J_{x}$-vertex,
there is an extra factor $-i$ $(i)$ for both $\Gamma_{\pm}(\omega)$.

  * •

If an electrical field line (incoming or outgoing) couples to a
$J_{y}$-vertex, this gives a factor $\mp$ 1 for $\Gamma_{\pm}(\omega)$.

Application of these rules directly leads to Eq. (23) in the non-interacting
case. For the integrated impurity depletion rate, let us consider for instance
the contribution of the two diagrams of Fig. 9(c), 9(d), to be denoted $D$. It
reads

$\displaystyle
D=-n_{\downarrow}g^{2}E^{2}A_{0}\int_{0}^{\infty}d\omega\int\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}\frac{d{\boldsymbol{q}}}{(2\pi)^{2}}\
\text{Im}\
\bigg{\\{}\int\frac{d\omega_{k}}{2\pi}\int\frac{d\omega_{q}}{2\pi}\left(-2iJ^{y}_{\uparrow,21}({\boldsymbol{k}})J^{x}_{\downarrow}({\boldsymbol{q}})W^{2}+2iJ^{y}_{\uparrow,12}({\boldsymbol{k}})J^{x}_{\downarrow}({\boldsymbol{q}})\overline{W}^{2}\right)\frac{1}{\omega^{2}}$
(49)
$\displaystyle\frac{1}{\omega_{q}-{\epsilon_{\downarrow}}({\boldsymbol{q}})+i0^{+}}\frac{1}{\omega_{q}-\omega-{\epsilon_{\downarrow}}({\boldsymbol{q}})+i0^{+}}\frac{1}{\omega_{k}-{\epsilon_{\uparrow,1}}({\boldsymbol{k}})-i0^{+}}\frac{1}{\omega+\omega_{k}-{\epsilon_{\uparrow,2}}({\boldsymbol{k}})+i0^{+}}\frac{1}{\omega+\omega_{k}-\omega_{q}-{\epsilon_{\uparrow,2}}({\boldsymbol{k}}-{\boldsymbol{q}})+i0^{+}}\bigg{\\}}\
.$

Here $W$ is shorthand for the proper interaction matrices (cf. Eq. (10)). The
third propagator is advanced (it corresponds to a majority hole) and has a
$-i0^{+}$ term in the denominator, the other propagators are retarded.
Performing the $\omega_{k},\omega_{q}$ integrals yields

$\displaystyle D=-n_{\downarrow}g^{2}E^{2}A_{0}\
\int\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}\frac{d{\boldsymbol{q}}}{(2\pi)^{2}}\int_{>0}d\omega\
\text{Im}\
\bigg{\\{}\left(-2iJ^{y}_{\uparrow,21}({\boldsymbol{k}})J^{x}_{\downarrow}({\boldsymbol{q}})W^{2}+2iJ^{y}_{\uparrow,12}({\boldsymbol{k}})J^{x}_{\downarrow}({\boldsymbol{q}})\overline{W}^{2}\right)\frac{1}{\omega^{2}}$
(50)
$\displaystyle\frac{1}{\omega+{\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}}-{\boldsymbol{q}})-{\epsilon_{\downarrow}}({\boldsymbol{q}})+i0^{+}}\frac{1}{{\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\downarrow}}({\boldsymbol{q}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}}-{\boldsymbol{q}})+i0^{+}}\frac{1}{\omega+{\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}})+i0^{+}}\bigg{\\}}\
.$

The expression involving the currents is real, and the imaginary part comes
from the propagators only. They yield a sum of two delta-functions, since the
propagator in the middle is real. Computing the $\omega$-integral, after some
trivial algebra one then finds

$\displaystyle D=2\pi
E^{2}A_{0}\cdot-2g^{2}n_{\downarrow}\int\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}\frac{d{\boldsymbol{q}}}{(2\pi)^{2}}\
\text{Im}\left\\{J^{y}_{\uparrow,12}({\boldsymbol{k}})J_{\downarrow}^{x}({\boldsymbol{q}})W^{2}\right\\}\frac{2{\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}}-{\boldsymbol{q}})-{\epsilon_{\downarrow}}({\boldsymbol{q}})}{({\epsilon_{\uparrow,2}}({\boldsymbol{k}})-{\epsilon_{\uparrow,1}}({\boldsymbol{k}}))^{2}({\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}}-{\boldsymbol{q}})-{\epsilon_{\downarrow}}({\boldsymbol{q}}))^{3}}\
,$ (51)

which is precisely $2\pi E^{2}A_{0}$ times the
${\sigma_{\downarrow\uparrow}}$-contribution of the “direct” diagram, cf.
(10). Evaluation of the other non-vanishing drag diagrams (crossed diagram and
diagrams with $J_{\downarrow}^{y},J_{\uparrow}^{x}$ interchanged) proceeds in
the same manner.

Since diagrams where both external field lines couple to the impurity vanish
when forming $\Delta\Gamma_{\downarrow}$, the only remaining non-zero diagrams
are those of Fig. 9(a), 9(b) plus those with inverted directions of the
external field lines. After some straightforward simplifications, one finds a
total contribution

$\displaystyle\frac{n_{\downarrow}g^{2}}{(2\pi)^{4}}4E^{2}A_{0}\
\int_{0}^{\infty}\frac{d\omega}{\omega^{2}}\int
d{\boldsymbol{k}}d{\boldsymbol{q}}\
\text{Im}\left[J_{\uparrow,21}^{x}({\boldsymbol{k}}-{\boldsymbol{q}})J^{y}_{\uparrow,12}({\boldsymbol{k}})W_{11}({\boldsymbol{k}},-{\boldsymbol{q}},-{\boldsymbol{q}})W_{22}({\boldsymbol{k}}-{\boldsymbol{q}},{\boldsymbol{{0}}},{\boldsymbol{q}})\right]$
(52)
$\displaystyle\text{Im}\bigg{\\{}(-1)\frac{1}{-\omega+{\epsilon_{\uparrow,2}}({\boldsymbol{k}}-{\boldsymbol{q}})-{\epsilon_{\uparrow,1}}({\boldsymbol{k}}-{\boldsymbol{q}})-i0^{+}}\frac{1}{\omega+{\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}})+i0^{+}}$
$\displaystyle\cdot\left(\frac{1}{\omega-{\epsilon_{\uparrow,2}}({\boldsymbol{k}})+{\epsilon_{\uparrow,1}}({\boldsymbol{k}}-{\boldsymbol{q}})-{\epsilon_{\downarrow}}({\boldsymbol{q}})+i0^{+}}+\frac{1}{\omega+{\epsilon_{\uparrow,1}}({\boldsymbol{k}})-{\epsilon_{\uparrow,2}}({\boldsymbol{k}}-{\boldsymbol{q}})-{\epsilon_{\downarrow}}({\boldsymbol{q}})+i0^{+}}\right)\bigg{\\}}\
.$

It is readily seen that this expression is invariant under
${\epsilon_{\uparrow,1}}\leftrightarrow-{\epsilon_{\uparrow,2}}$, which
implies the particle-hole symmetry claimed in the main text. We have also
checked this symmetry explicitly for the Haldane model by numerically
implementing Eq. (52).

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