# Quantized Nonlinear Conductance in Ballistic Metals

C. L. Kane  Department of Physics and Astronomy, University of Pennsylvania,
Philadelphia, PA 19104

###### Abstract

We introduce a non-linear frequency dependent $D+1$ terminal conductance that
characterizes a $D$ dimensional Fermi gas, generalizing the Landauer
conductance in $D=1$. For a ballistic conductor we show that this conductance
is quantized and probes the Euler characteristic of the Fermi sea. We
critically address the roles of electrical contacts and of Fermi liquid
interactions, and we propose experiments on 2D Dirac materials such as
graphene using a triple point contact geometry.

A dramatic consequence of the role of topology in the structure of quantum
matter is the existence of topological invariants that are reflected in
quantized response functions. The hallmark of this is the integer quantized
Hall effect (IQHE)Klitzing  _et al._ (1980), which probes the Chern number
characterizing the topology of a gapped 2 dimensional (2D) electronic
phaseThouless  _et al._ (1982). Quantum topology also plays a role in the
electrical response of metals. For example, the Berry phase associated with
the Fermi surface of a 2D metal contributes to the non-quantized part of the
anomalous Hall conductivityHaldane (2004). In 3D, the Chern number associated
with the Fermi surface in a Weyl semimetal leads to a quantized circular
photogalvanic effectde Juan  _et al._ (2017) in the absence of disorder and
interactionsAvdoshkin  _et al._ (2020). In addition to the quantum topology
associated with the twisting of the quantum states on the Fermi surface,
metals also exhibit a simpler geometric topology associated with the shape of
the Fermi surface. It is well known that noble metals, like copper, have a
Fermi surface with a non-trivial genusAshcroft and Mermin (1976). While Fermi
surfaces have been mapped in detail, and Lifshitz transitionsLifshitz (1960)
where their topology changes have been characterizedVolovik (2017, 2018),
Fermi surface topology has not been measured directly. Here we pose the
question of whether the topology of the Fermi surface is associated with a
quantized response.

An indication that the answer is affirmative is provided by 1D. The Landauer
conductance of a ballistic 1D conductor is $e^{2}/h$ times the number of
occupied bandsLandauer (1957); Fisher and Lee (1981). While this quantization
is related to the IQHE, there are important differences. First it is less
robust, since it relies on reflectionless contacts and the absence of
scattering. Nonetheless, conductance quantization has been observed in quantum
point contactsvan Wees  _et al._ (1988), 1D semiconductor wiresHonda  _et al._
(1995); van Weperen  _et al._ (2013) and carbon nanotubesFrank  _et al._
(1998), albeit with less precision than the IQHE. A second difference is that
unlike the IQHE, the quantized value does not reflect the topology of a 2D
gapped state, but rather the topology of the 1D filled Fermi sea.

In this paper, we generalize this notion to higher dimensions. For a $D$
dimensional ballistic conductor with suitably defined ideal leads, we
introduce a $D+1$ terminal frequency dependent nonlinear conductance,

$I_{D+1}(\omega_{\Sigma})=G(\\{\omega_{p}\\})\prod_{p=1}^{D}V_{p}(\omega_{p}),$
(1)

where $I_{p}(\omega_{p})$ ($V_{p}(\omega_{p})$) are the current (voltage) in
lead $p$ at frequency $\omega_{p}$ and $\omega_{\Sigma}=\sum_{p}\omega_{p}$.
We will argue that for low (but not too low) frequency, $G(\\{\omega_{p}\\})$
has the form,

$G(\\{\omega_{p}\\})=\frac{i\omega_{\Sigma}}{\prod_{p}(i\omega_{p})}\frac{e^{D+1}}{h^{D}}\chi_{F},$
(2)

where $\chi_{F}$ is the Euler characteristic of the $D$-dimensional Fermi sea.
We will establish this for non-interacting electrons by first presenting a
simple thought experiment, which is formalized by a semi-classical Boltzmann
transport theory. This will be followed by a more general quantum non-linear
response theory, which reproduces the Boltzmann theory. We will focus mainly
on the case $D=2$, and will critically assess the prospects for experimentally
measuring $\chi_{F}$ in a 2D conductor using a triple point contact. Crucial
issues to be addressed include the role of electrical contacts and electron-
electron interactions, which place bounds on the applicability of (2).

The Euler characteristic is defined asNakahara (1990)

$\chi_{F}=\sum_{l=0}^{D}(-1)^{l}b_{l},$ (3)

where $b_{l}$ is the $l$’th Betti number, given by the rank of the $l$’th
homology group, which counts the topologically distinct $l$-cycles. In 1D,
$\chi_{F}$ is the number of disconnected components of the Fermi sea. In
general, $\chi_{F}$ can be expressed as a sum over the disconnected components
of the Fermi surface. In 2D, electron-like, hole-like and open Fermi surfaces
contribute $+1$, $-1$ and $0$, respectively. In 3D, each Fermi surface with
genus $g_{k}$ contributes $1-g_{k}$. Note that completely empty bands and
completely filled bands both have $\chi_{F}=0$, and electron-like and hole-
like Fermi surfaces have opposite sign for even $D$.

Morse theoryMilnor (1969); Nash and Sen (1988) provides a representation of
$\chi_{F}$ in terms of the critical points of the electronic dispersion
$E({\bf k})$, where ${\bf v}_{\bf k}=\nabla_{\bf k}E({\bf k})/\hbar=0$ for
$E({\bf k})<E_{F}$:

$\chi_{F}=\sum_{m}\eta_{m}.$ (4)

$m$ labels the critical points (assumed non-degenerate) with signature
$\eta_{m}={\rm sgn}({\rm det}[\partial^{2}E({\bf k}_{m})/\partial
k_{i}\partial k_{j}])$. This shows that $\chi_{F}$ changes at a Lifshitz
transitionLifshitz (1960), when a minimum, maximum or saddle point passes
through $E_{F}$, signaling a change in Fermi surface topology.

To motivate our result, we review and then generalize a thought
experimentLaughlin (1981) that explains the quantization of the 1D Landauer
conductance. Consider an infinitely long 1D electron gas (1DEG), with
electronic states $E(k)=\hbar^{2}k^{2}/2m$ filled to $E_{F}$. Apply a $h/e$
voltage pulse $V(t)$ by introducing a slowly varying electric field $E(x,t)$
that is nonzero near $x=0$ and $t=0$, such that $\int dxdtE(x,t)=h/e$. This
will lead to a charge $Q=Gh/e$ transferred into the right lead, where $G$ is
the conductance. The charge $Q=e$ can be deduced from the fact that the
impulse $V(t)$ transfers precisely one electron between the left- and right-
moving Fermi points, reflecting the chiral anomaly associated with 1D chiral
fermions. The chiral anomaly is a consequence of the fact that the right and
left movers are connected at the critical point $k=0$. Due to the impulse one
electron crosses $k=0$ and changes direction. This argument can be generalized
to a more complicated dispersion $E(k)$. An electron will change direction at
every critical point inside the Fermi sea, leading to a net transferred charge
$Q=e\chi_{F}$, with $\chi_{F}$ given in (4). It follows that
$G=\chi_{F}e^{2}/h$.

We now seek a version of this argument for higher $D$. Consider a 2DEG with
dispersion $E({\bf k})$ defined on an infinite plane that is divided into 3
regions that meet at a point. Apply a $h/e$ pulse $V_{1}(t)$ to region $1$,
followed by a $h/e$ pulse $V_{2}(t)$ to region $2$ by introducing electric
fields near their boundaries. Each pulse will lead to a charge transferred to
lead 3 that scales with the length of the contact. However, we will argue that
the excess charge $Q_{3}$, defined as the charge transferred due to the two
pulses with the charge transferred for independent pulses subtracted off, will
be universal and given by $Q_{3}=e\chi_{F}$, where $\chi_{F}$ is the Euler
characteristic of the 2D Fermi sea. This excess charge defines a second order
non-linear response that can be isolated in the frequency domain.

Figure 1: (a,b) A thought experiment in which $h/e$ voltage pulses are applied
to regions 1 and 2 in (a). (c) shows a hypothetical 2D Fermi sea with
$\chi_{F}=-1$. After pulse $V_{1}(t)$ for $x>0$ there are extra electrons
(shown in blue) propagating to the right. Pulse $V_{2}(t)$ accelerates those
electrons, and $v_{y}$ changes sign at the points indicated by the red dots in
a direction indicated by the arrows. The net excess charge in region 3 is
determined by the difference between the number of concave and convex critical
points on the Fermi surface with ${\bf v}_{\bf k}\propto+\hat{x}$, which
measures $\chi_{F}$.

It is simplest to consider the geometry in Fig. 1a, where region 1 is the
half-plane $x<0$, and regions 2 (3) are the quadrants $x>0$, $y<0$($y>0$). In
that case, the $V_{1}(t)$ pulse accelerates electrons in the $x$ direction, so
that for every value of $k_{y}$ on the Fermi surface there is one extra
electron propagating to the right ($v_{x}>0$) for $x>0$, and one extra hole
propagating to the left ($v_{x}<0$) for $x<0$. $Q_{3}$ will be determined by
the effect of the $V_{2}(t)$ pulse on those extra electrons with $x>0$, which
are accelerated in the $y$ direction. As in the 1D example discussed above,
the transferred charge can be determined by counting those electrons that
change directions at the critical points on the Fermi surface where $v_{x}>0$
and $v_{y}=0$. Referring to the hypothetical Fermi surface in Fig. 1c, these
arise at the points indicated by the dots, which come in two varieties
distinguished by whether the Fermi surface is concave (convex) with
$\partial^{2}E/\partial k_{y}^{2}>0$($<0$). $Q_{3}$ is then $e$ times the sum
over those critical points with signs ${\rm sgn}[\partial^{2}E/\partial
k_{y}^{2}]$. By inspecting Fig. 1c, it is clear that this is $\chi_{F}$. This
will be proven below. We conclude that $Q_{3}=e\chi_{F}$.

The above argument can be sharpened by developing a semi-classical Boltzmann
transport theory. In the absence of scattering the electron distribution
function $f({\bf k},{\bf r},t)$ satisfies the collisionless Boltzmann
equation,

$(\partial/\partial t+{\bf v}_{\bf k}\cdot\nabla_{\bf r}+e{\bf
E}\cdot\nabla_{\bf k}/\hbar)f=0.$ (5)

Consider two weak pulses, ${\bf
E}_{1}(x_{1},y_{1},t)=\xi_{1}\delta(t-t_{1})\delta(x_{1})\hat{x}$ and ${\bf
E}_{2}(x_{2},y_{2},t)=\xi_{2}\delta(t-t_{2})[\delta(y_{2})\theta(x_{2})\hat{y}-\delta(x_{2})\theta(-y_{2})\hat{x}]$.
We compute the charge

$Q_{3}(t_{3})=e\int\frac{d^{2}k}{(2\pi)^{2}}\int_{0}^{\infty}dx_{3}dy_{3}\delta
f({\bf k},x_{3},y_{3},t_{3})$ (6)

perturbatively at order $\xi_{1}\xi_{2}$ for $t_{1}<t_{2}<t_{3}$. Integrating
(5) to this order gives $\delta
f=e^{2}\xi_{1}\xi_{2}\delta\tilde{f}/\hbar^{2}$ with

$\delta\tilde{f}=\int dy_{1}dx_{2}\delta({\bf r}_{32}-{\bf v}_{\bf
k}t_{32})\frac{\partial}{\partial{k_{y}}}[\delta({\bf r}_{21}-{\bf v}_{{\bf
k}}t_{21})\frac{\partial f_{0}}{\partial{k_{x}}}],$ (7)

where $f_{0}({\bf k})=\theta(E_{F}-E({\bf k}))$, $t_{ij}=t_{i}-t_{j}$, ${\bf
r}_{32}=(x_{3}-x_{2},y_{3})$, ${\bf r}_{21}=(x_{2},-y_{1})$, and $x_{2}$
($y_{1}$) are integrated from $0$ ($-\infty$) to $\infty$. The second term in
${\bf E}_{2}$, with $x_{2}=0$, is absent because the $\delta$ functions can
not be satisfied. After plugging (7) into (6), the four spatial integrals
cancel the $\delta$ functions, but since $x_{2}>0$ ($y_{3}>0$), we require
$v_{x}>0$ ($v_{y}>0$) inside (outside) $\partial/\partial k_{y}$. After
integrating by parts on $k_{y}$ and replacing $\xi_{1,2}\rightarrow h/e$ we
obtain

$Q_{3}=e\int d^{2}k\frac{\partial f_{0}({\bf k})}{\partial
k_{x}}\theta(v_{x}({\bf k}))\frac{\partial\theta(v_{y}({\bf k}))}{\partial
k_{y}}.$ (8)

This captures the result of the heuristic argument above: $\partial
f_{0}/\partial k_{x}$ isolates the Fermi surface, while
$\theta(v_{x})\partial(\theta(v_{y}))/\partial k_{y}$ isolates the critical
points on the Fermi surface identified in Fig. 1c. To make contact with Eq. 4,
it is convenient to add zero to the integrand in the form $-(\partial
f_{0}/\partial k_{y})\theta(v_{x})\partial\theta(v_{y})/\partial k_{x}$. This
is clearly zero, since $\partial f_{0}/\partial k_{y}=v_{y}\partial
f_{0}/\partial E$, and $\partial\theta(v_{y})/\partial k_{x}$ fixes $v_{y}=0$.
This allows us to integrate by parts to obtain,

$Q_{3}=e\int d^{2}kf_{0}({\bf k})[\frac{\partial\theta(v_{x})}{\partial
k_{x}}\frac{\partial\theta(v_{y})}{\partial
k_{y}}-\frac{\partial\theta(v_{x})}{\partial
k_{y}}\frac{\partial\theta(v_{y})}{\partial k_{x}}]$ (9)

The integrand is only non-zero near critical points where $v_{x}=v_{y}=0$. The
integral evaluates the signature of each critical point, leading to

$Q_{3}=e\sum_{m}\eta_{m}=e\chi_{F}.$ (10)

A 3D version of this argument can be formulated by applying voltage pulses to
three regions $(z<0)$, $(z>0,x<0)$ and $(z>0,x>0,y<0)$ and determining the
charge $Q_{4}$ in $(z>0,x>0,y>0)$, leading to $Q_{4}=e\chi_{F}$, where
$\chi_{F}$ characterizes the 3D Fermi sea.

The above argument suggests that in the frequency domain, the non-linear
response is given by (2). The Boltzmann theory can be extended to the
frequency domain, and generalized to show that (2) is independent of the
angles between the boundaries of the regions as well as the detailed spatial
profile of the fields, provided they are localized near the boundaries. Before
doing that, however, we will formulate a more general quantum non-linear
response theory, and show the Boltzmann theory follows, provided the applied
fields vary slowly in space and time. To this end, we introduce the
Hamiltonian

${\cal H}={\cal
H}_{0}+(V_{1}\hat{Q}_{1}e^{(\eta-i\omega_{1})t}+V_{2}\hat{Q}_{2}e^{(\eta-i\omega_{2})t}+{\rm
h.c.})$ (11)

with ${\cal H}_{0}=\sum_{\bf k}E({\bf k})c_{\bf k}^{\dagger}c_{\bf k}$ and
$\hat{Q}_{p}=\int d{\bf r}Q_{p}({\bf r})\rho({\bf r})$ is defined in terms of
the density operator $\rho({\bf r})$ for each of the three regions in Fig. 1a.
$Q_{p}({\bf r})$ is $1$ ($0$) inside (outside) region $p$ and is assumed to
transition smoothly between $1$ and $0$ in a width $b$ near the boundary, with
$k_{F}b\gg 1$.

Figure 2: Feynman diagrams for the conductance. (a,b) describe the 2D non-
linear conductance and (c) describes the 1D linear conductance in the absence
of interactions and give the quantized conductance determined by $\chi_{F}$.
(d) and (e) show corrections to (a,b) and (c) that describe screening due to
electron interactions and modify the quantized result.

We compute the charge $Q_{3}(t)$ at frequency $\omega_{1}+\omega_{2}$ to order
$V_{1}V_{2}$. We adopt a scalar potential formulation, which avoids the
diamagnetic term. The response has structure similar to second order nonlinear
optical responseKraut and von Baltz (1979); von Baltz and Kraut (1981); Zhang
_et al._ (2018), and is determined by evaluating the Feynman diagrams in Fig.
2a,b.

$Q_{3}(\omega_{1}+\omega_{2})=\alpha(\omega_{1},\omega_{2})V_{1}V_{2}$ (12)

with

$\displaystyle\alpha(\omega_{1},\omega_{2})=$
$\displaystyle\frac{e^{3}}{\hbar^{2}}\sum_{l,m,n}\frac{f_{l}-f_{m}}{\omega_{1}-\omega_{lm}+i\eta}[\frac{Q_{1}^{lm}Q_{2}^{mn}Q_{3}^{nl}}{\omega_{1}+\omega_{2}-\omega_{ln}+i\eta}$
$\displaystyle-\frac{Q_{1}^{lm}Q_{3}^{mn}Q_{2}^{nl}}{\omega_{1}+\omega_{2}-\omega_{nm}+i\eta}]+(1\leftrightarrow
2).$ (13)

Here $l,m,n$ label momenta, $\omega_{lm}=(E({\bf k}_{l})-E({\bf
k}_{m}))/\hbar$, $f_{l}=f_{0}({\bf k}_{l})$ is a Fermi function, and

$Q_{p}^{lm}=\langle{\bf k}_{l}|\hat{Q}_{p}|{\bf k}_{m}\rangle=\int d{\bf
r}_{p}Q_{p}({\bf r}_{p})e^{i{\bf q}_{lm}\cdot{\bf r}_{p}}$ (14)

with ${\bf q}_{lm}={\bf k}_{l}-{\bf k}_{m}$. Since $Q_{p}({\bf r})$ varies on
the scale of $b$, an expansion in ${\bf q}_{lm}$ and ${\bf q}_{nm}$ is
justified. As shown in supplemental section A, the ${\bf q}_{lm}$ and ${\bf
q}_{nm}$ integrals can be performed to obtain,

$\displaystyle\alpha=$ $\displaystyle\int\frac{d^{2}{\bf k}d^{6}{\bf
r}_{1,2,3}}{(2\pi)^{2}}\nabla^{a}_{\bf r}Q_{1}({\bf r}_{1})\nabla^{b}_{\bf
r}Q_{2}({\bf r}_{2})Q_{3}({\bf r}_{3})[\nabla_{\bf k}^{a}f_{0}({\bf k})$
$\displaystyle D({\bf r}_{21},{\bf k},\omega_{1})\nabla_{\bf k}^{b}D({\bf
r}_{32},{\bf k},\omega_{1}+\omega_{2})]+(1\leftrightarrow 2)$ (15)

with $D({\bf r},{\bf k},\omega)=e^{-(\eta-i\omega)|{\bf r}|/|{\bf v}_{\bf
k}|}\delta({\bf r}\times{\bf v}_{\bf k})\theta({\bf r}\cdot{\bf v}_{\bf k})$.
This form of the response also follows from solving (5) in the frequency
domain with ${\bf E}_{p}({\bf r},t)=-\nabla Q_{p}({\bf
r})V_{p}e^{(\eta-i\omega_{p})t}$.

In supplemental section B, we evaluate (15) for the infinite plane in which
three rays separate regions that subtend angles $\varphi_{p}$ (see Fig. 3).
Provided $\omega_{p}\ll v_{F}/b$, the result is independent of the detailed
potential profile on the boundaries of width $b$. Provided all angles
$\varphi_{p}<\pi$, the non-linear conductance is independent of $\varphi_{p}$
and given by (2). This rigorous result, which is exact for the infinite plane
and non-interacting electrons is the central result of this paper. To address
experimental feasibility we must consider the role of contacts as well as
electron-electron interactions. These both introduce complications into the
analysis.

Figure 3: A triple point contact as a model experimental geometry. A 2D
electron gas is connected to three metallic leads of size larger than the
tunneling mean free path $\ell$. The leads are separated by $b$ and subtend
angles $\varphi_{1,2,3}$.

As a model for electrical contacts, consider Fig. 3, which depicts a 2D
electron gas (2DEG) of size $L$ with large area tunnel contacts to ideal
metallic leads separated by a distance $b$. Provided the capacitance between
the contacts and the 2DEG is sufficiently large, the voltages in the leads
will establish a potential profile in the 2DEG as in Eq. 11. We assume the
tunnel barrier is in a regime in which the mean free path $\ell$ for tunneling
from the 2DEG to the leads satisfies $k_{F}^{-1}\ll\ell\ll L$. This defines a
dwell time $\tau=\ell/v_{F}$ for electrons in the 2DEG. In the pulse
construction, we clearly require $t_{21}<\tau$, since for $t_{21}>\tau$ the
first pulse has disappeared before the second arrives. In the frequency domain
calculation, tunneling to the leads introduces an exponential decay $e^{-|{\bf
r}|/\ell}$ into $D({\bf r},{\bf k},\omega)$, which cuts off the small
$\omega_{1}$ divergence in $\alpha$, effectively replacing
$\omega_{1}\rightarrow\omega_{1}+i/\tau$. The coupling to the leads therefore
places a lower bound $\omega_{1},\omega_{2}\gg\omega_{c}$ for the
applicability of (2) with $\omega_{c}\sim\tau^{-1}$ . Note, however, that
there is no bound on $\omega_{\Sigma}=\omega_{1}+\omega_{2}$, and below we
will see that it is advantageous to consider $\omega_{\Sigma}\ll\omega_{c}$.

A second complication involves the role of electron-electron interactions. In
1D, the analog of our calculation is the Kubo formula conductance $G_{\rm
Kubo}$, which is given by $G_{\rm Kubo}=Ke^{2}/h$, where $K<1$ is the
Luttinger parameter characterizing repulsive electron interactionsApel and
Rice (1982); Kane and Fisher (1992). However, $G_{\rm Kubo}$ does not
correctly account for the electrical contacts, and it was argued that for
Fermi liquid leads on a 1DEG of length $L$, $G(\omega)=e^{2}/h$ for $\omega\ll
v_{F}/L$Maslov and Stone (1995); Ponomarenko (1995). An appealing
interpretation of this was explained by KawabataKawabata (1996), who argued
that $G_{\rm Kubo}$ computed in an infinite system is renormalized because it
describes the response to the applied voltage, while the quantized
conductance, which reflects the chiral anomaly, is the response to the self-
consistent potential, which includes screening due to interactions. Moreover,
for the DC conductance it is the self-consistent potential that determines the
measured electrochemical potential difference. ShimizuShimizu (1996)
emphasized the similarity of this description to Fermi liquid theoryPines and
Nozieres (1966). In terms of Feynman diagrams, for the response to the applied
field, the Kubo formula bubble diagram (Fig. 2c) is dressed by convolving with
RPA-like polarization bubbles (Fig. 2d), $\Pi(x,\omega)$, represented here in
real space. In contrast, response to the self-consistent potential does not
include that dressing. For a 1DEG in which leads are modeled by setting the
interactions to zero for $|x|>L$, it can be checked that since
$\Pi(x,\omega)\sim(\omega/v_{F})e^{i\omega x/v_{F}}$, and $|x|<L$ the
interaction corrections vanish for $\omega\ll v_{F}/L$.

To incorporate electron interactions into the calculation of the 2D non-linear
response we adopt a renormalized Fermi liquid descriptionShankar (1994) in
which quasiparticles near $E_{F}$ interact with an energy $f_{{\bf k}{\bf
k}^{\prime}}n_{\bf k}n_{{\bf k}^{\prime}}$, where $n_{\bf k}=c_{\bf
k}^{\dagger}c_{\bf k}$. At low energy, the important interaction corrections
involve RPA bubbles, and summing the diagrams like 2(e) is equivalent to
incorporating the Fermi liquid interactions $f_{{\bf k}{\bf k}^{\prime}}$ into
the Boltzmann equationPines and Nozieres (1966); Shankar (1994). This is
difficult to solve in general, but by evaluating the simplest diagram at first
order in $f_{{\bf k}{\bf k^{\prime}}}$ it can be checked that the response is
modified: $\chi_{F}\rightarrow\chi_{F}+O(N(E_{F})f)$, where $f$ is an average
of $f_{{\bf k}{\bf k}^{\prime}}$ over the Fermi surface. Thus, the non-linear
response in 2D is modified by the Fermi liquid parameters just as the linear
response in 1D is modified by the Luttinger parameter. For
$|\omega_{\Sigma}|=|\omega_{1}+\omega_{2}|\ll\omega_{c}$, the bubble
$\Pi(\omega_{\Sigma},{\bf
r})\propto(\omega_{\Sigma}/v_{F})e^{i\omega_{\Sigma}|{\bf r}|/v_{F}}$ vanishes
for $\omega_{\Sigma}\ll\omega_{c}$ as for $D=1$. However, since we must
consider $\omega_{1},\omega_{2}\gg\omega_{c}$, the interaction corrections
remain for the $Q_{1}$ and $Q_{2}$ vertices. The origin of the correction is
the same as in $D=1$: due to interactions, the potential is screened, as is
accounted for by the RPA bubbles. The response to the self-consistent
potential is given by the diagrams in Fig. 2a,b.

At finite frequency, more detailed modeling is required to determine the
relation between the self-consistent potential and the measured voltage. In
the absence of that, it will be fruitful to consider weakly interacting
systems. Consider a 2D Dirac material, such as graphene, with density of
states $N(E)\sim|E|/v_{F}^{2}$. For a short ranged interaction (screened by
the leads) $N(E_{F})f\ll 1$ for sufficiently small $E_{F}$. Interestingly, the
Fermi surface is electron-like (hole-like) for $E_{F}>0$($E_{F}<0$), so the
response characterized by $\chi_{F}=4{\rm sgn}E_{F}$ (including spin and
valley) changes sign at charge neutrality.

Our analysis opens several avenues for further inquiry. On the practical side,
it will be interesting to search for other measurable quantities that probe
$\chi_{F}$. Promising candidates include low frequency current noise as well
as thermal conductance. It will also be interesting to explore ways in which
$\chi_{F}$ provides a fundamental characterization of a degenerate Fermi gas.
Our analysis suggests that $\chi_{F}$ defines a kind of “higher order”
anomaly. In 1D, the chiral anomaly characterizes the connection between left
and right moving electrons, which leads to a lack of conservation of the right
movers. For $D>1$, $\chi_{F}$ characterizes a more general violation of
conservation due to the fact that electrons propagating in different
directions are connected at critical points ${\bf k}_{m}$. This is related to
the Fermi surface anomaly discussed in Ref. Else  _et al._ , 2021, which
characterizes a particular point on the Fermi surface. However, unlike that
description, $\chi_{F}$ provides a global characterization of the Fermi
surface. For $D=1$ the bipartite entanglement entropy has a universal term
$S=(c/3){\rm log}L$Holzhey  _et al._ (1994); Calabrese and Cardy (2004) with
$c=\chi_{F}$. This has been generalized to higher $D$, where $S$ describes a
logarithmic area law entanglement that involves the projected area of the
Fermi surfaceGioev and Klich (2006); Swingle (2010), which is non-zero even
for a system of decoupled 1D wiresDing  _et al._ (2012), with $\chi_{F}=0$. We
speculate that for $D>1$, $\chi_{F}$ shows up in an intrinsically $D$
dimensional entanglement measure.

We thank Patrick Lee and Pok Man Tam for helpful suggestions. This work was
supported by a Simons Investigator Grant from the Simons Foundation.

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Supplementary Material

##  Appendix A Non-linear response for slowly varying fields

In this section we evaluate the quantum non-linear response formula, Eq. 13
for potentials that vary slowly in space and time, and show that the result
agrees with the perturbative solution of the Boltzmann equation, Eq. 5. In the
slowly varying limit, all of the transitions in (13) will be intra-band, so a
single band theory, characterized by dispersion $E({\bf k})$ will suffice.

Our starting point is to write Eq. 13 in real space.

$\alpha(\omega_{1},\omega_{2})=\frac{e^{3}}{\hbar^{2}}\int d^{6}{\bf
r}_{1,2,3}Q_{1}({\bf r}_{1})Q_{2}({\bf r}_{2})Q_{3}({\bf
r}_{3})\chi(\left\\{\omega_{p},{\bf r}_{p}\right\\})$ (16)

with

$\chi(\left\\{{\bf r}_{p},\omega_{p}\right\\})=\sum_{{\bf
k}_{l,m,n}}\frac{f_{l}-f_{m}}{\omega_{1}-\omega_{lm}+i\eta}[\frac{\rho_{lm}({\bf
r}_{1})\rho_{mn}({\bf r}_{2})\rho_{nl}({\bf
r}_{3})}{\omega_{1}+\omega_{2}-\omega_{ln}+2i\eta}-\frac{\rho_{lm}({\bf
r}_{1})\rho_{mn}({\bf r}_{3})\rho_{nl}({\bf
r}_{2})}{\omega_{1}+\omega_{2}-\omega_{nm}+2i\eta}]+(1\leftrightarrow 2).$
(17)

When $Q_{p}({\bf r}_{p})$ is slowly varying, the matrix elements of the
density operator will be dominated by intraband terms with small momenta,
${\bf q}_{lm}={\bf k}_{l}-{\bf k}_{m}\sim 1/b$,

$\rho_{lm}({\bf r})=\langle{\bf k}_{l}|\rho({\bf r})|{\bf
k}_{m}\rangle=e^{-i{\bf q}_{lm}\cdot{\bf r}}(1+O({\bf q}_{lm})).$ (18)

Expanding to leading order in ${\bf q}_{lm}$ then gives

$\chi(\left\\{{\bf r}_{p},\omega_{p}\right\\})=\sum_{{\bf
k}_{l,n,m}}\frac{f_{l}-f_{m}}{\omega_{1}-{\bf q}_{lm}\cdot{\bf
v}_{lm}+i\eta}[\frac{e^{i({\bf q}_{lm}\cdot{\bf r}_{31}+{\bf q}_{nm}\cdot{\bf
r}_{32})}}{\omega_{1}+\omega_{2}-{\bf q}_{ln}\cdot{\bf
v}_{ln}+2i\eta}-\frac{e^{i({\bf q}_{lm}\cdot{\bf r}_{31}+{\bf q}_{ln}\cdot{\bf
r}_{32})}}{\omega_{1}+\omega_{2}-{\bf q}_{nm}\cdot{\bf
v}_{nm}+2i\eta}]+(1\leftrightarrow 2)$ (19)

where since it is necessary to keep order $q^{2}$ terms in the denominator we
write $\omega_{lm}={\bf q}_{lm}\cdot{\bf v}_{lm}+O(q^{3})$, with ${\bf
v}_{lm}={\bf v}_{({\bf k}_{l}+{\bf k}_{m})/2}$. If we define ${\bf k}=({\bf
k}_{l}+{\bf k}_{m})/2$, then ${\bf v}_{lm}={\bf v}_{\bf k}$, ${\bf
v}_{ln}={\bf v}_{{\bf k}+{\bf q}_{nm}/2}$ and ${\bf v}_{nm}={\bf v}_{{\bf
k}-{\bf q}_{ln}/2}$. We next define ${\bf q}_{1}={\bf q}_{lm}$ and ${\bf
q}_{2}={\bf q}_{mn}$ (${\bf q}_{nl}$) in the first (second) term. It follows
that

$\displaystyle\chi(\left\\{{\bf r}_{p},\omega_{p}\right\\})=$
$\displaystyle\sum_{{\bf k},{\bf q}_{1},{\bf q}_{2}}e^{i({\bf q}_{1}\cdot{\bf
r}_{31}+{\bf q}_{2}\cdot{\bf r}_{32})}\frac{f({\bf k}+{\bf q}_{1}/2)-f({\bf
k}-{\bf q}_{1}/2)}{\omega_{1}-{\bf q}_{1}\cdot{\bf v}_{\bf k}+i\eta}$
$\displaystyle[\frac{1}{\omega_{1}+\omega_{2}-({\bf q}_{1}+{\bf
q}_{2})\cdot{\bf v}_{{\bf k}-{\bf
q}_{2}/2}+2i\eta}-\frac{1}{\omega_{1}+\omega_{2}-({\bf q}_{1}+{\bf
q}_{2})\cdot{\bf v}_{{\bf k}+{\bf q}_{2}/2}+2i\eta}]+(1\leftrightarrow 2)$
(20) $\displaystyle=$ $\displaystyle-\sum_{{\bf k},{\bf q}_{1},{\bf
q}_{2}}e^{i({\bf q}_{1}\cdot{\bf r}_{31}+{\bf q}_{2}\cdot{\bf
r}_{32})}\frac{{\bf q}_{1}\cdot\nabla_{\bf k}f({\bf k})}{\omega_{1}-{\bf
q}_{1}\cdot{\bf v}_{\bf k}+i\eta}{\bf q}_{2}\cdot\nabla_{\bf
k}\frac{1}{\omega_{1}+\omega_{2}-({\bf q}_{1}+{\bf q}_{2})\cdot{\bf v}_{{\bf
k}}+2i\eta}+(1\leftrightarrow 2)$ (21) $\displaystyle=$
$\displaystyle\nabla^{a}_{{\bf r}_{1}}\nabla^{b}_{{\bf r}_{2}}\sum_{{\bf
k},{\bf q}_{1},{\bf q}_{2}}e^{i({\bf q}_{1}\cdot{\bf r}_{31}+{\bf
q}_{2}\cdot{\bf r}_{32})}\frac{\nabla_{\bf k}^{a}f({\bf k})}{\omega_{1}-{\bf
q}_{1}\cdot{\bf v}_{\bf k}+i\eta}\nabla_{\bf
k}^{b}\frac{1}{\omega_{1}+\omega_{2}-({\bf q}_{1}+{\bf q}_{2})\cdot{\bf
v}_{{\bf k}}+2i\eta}+(1\leftrightarrow 2)$ (22)

Now we can do the sums over ${\bf q}_{1}$ and ${\bf q}_{2}$ using independent
variables ${\bf q}_{1}$ and ${\bf q}_{3}=-{\bf q}_{1}-{\bf q}_{2}$. Let us
define

$D({\bf r},\omega,{\bf v})=i\sum_{\bf q}\frac{e^{i{\bf q}\cdot{\bf
r}}}{\omega-{\bf q}\cdot{\bf v}+i\eta}=\frac{e^{(i\omega-\eta)|{\bf r}|/|{\bf
v}|}}{|{\bf v}|}\delta(\hat{\bf v}\times{\bf r})\theta(\hat{\bf v}\cdot{\bf
r})$ (23)

Then

$\chi(\left\\{{\bf r}_{p},\omega_{p}\right\\})=-\nabla^{a}_{{\bf
r}_{1}}\nabla^{b}_{{\bf r}_{2}}\sum_{{\bf k}}(\nabla^{a}_{\bf k}f({\bf
k}))D({\bf r}_{21},\omega_{1},{\bf v}_{\bf k})\nabla^{b}_{\bf k}D({\bf
r}_{32},\omega_{1}+\omega_{2},{\bf v}_{\bf k})+(1\leftrightarrow 2)$ (24)

Plugging (24) into (16) reproduces Eq. 15 in the main text. Note, that the
same result would follow by integrating the Boltzmann equation (Eq. 5) to
order $V_{1}V_{2}$, with ${\bf E}_{p=1,2}({\bf r},t)=-\nabla Q_{p}({\bf
r})V_{p}e^{(\eta-i\omega_{p})t}$.

##  Appendix B Non-linear response for a triple point contact

In this appendix we evaluate the non-linear response described by Eq. 15 for a
triple point contact, depicted in Fig. B1 in which the three contact regions
meet at a point and subtend arbitrary angles $\varphi_{p=1,2,3}$. We will show
that provided all three angles satisfy $\varphi_{p}<\pi$ the response has the
universal form characterized by Eq. 2. This generalizes the analysis of Eqs.
5-10 to the frequency domain and to a more general geometry. We will assume
the region boundaries, where $\nabla Q_{p}({\bf r})$ is non-zero are confined
a region of width $b$ about the straight rays shown in Fig. B1. The slowly
varying condition is satisfied provided $k_{F}b\gg 1$. Provided $\omega b\ll
1$, the result will be independent of $b$.

Due to the gradients on $Q_{p}({\bf r}_{p})$ for $p=1,2$ in Eq. 15, ${\bf
r}_{p=1,2}$ are confined to the boundary of region $p$, which consists of two
rays parametrized by $z_{p\pm}$, with normal $\hat{\varepsilon}_{p\pm}$. For
the first term (the one displayed in Fig. B1) only $z_{2+}$ will contribute
because for $z_{2-}$ there is no way to satisfy the delta-functions. For the
$(1\leftrightarrow 2)$ term only $z_{1+}$ will contribute.

Figure B1: A triple point contact with three contact regions that meet at a
point and subtend arbitrary angles $\varphi_{1,2,3}$. The boundary of region
$p$ consists of two rays along $\hat{n}_{p\sigma=\pm}$, with normal
$\hat{\varepsilon}_{p\sigma}$, and is parametrized by $z_{p\sigma}>0$.

The integral over ${\bf r}_{3}$ can be evaluated by considering

$\int d{\bf r}_{3}Q_{3}({\bf r}_{3})D({\bf r}_{32},\omega,{\bf
v})=\frac{\theta(\hat{\varepsilon}_{2+}\cdot{\bf v})}{\eta-i\omega}$ (25)

where we note that ${\bf r}_{2}$ is on the boundary of region 3, and the
$\delta$ and $\theta$ functions in D (Eq. 23) require ${\bf v}$ to have a
positive projection onto $\hat{\varepsilon}_{2+}$. We then get

$\alpha(\left\\{\omega_{p}\right\\})=\frac{e^{3}/\hbar^{2}}{2\eta-i(\omega_{1}+\omega_{2})}\sum_{{\bf
k},\sigma=\pm}\int_{0}^{\infty}dz_{1\sigma}dz_{2+}(\hat{\varepsilon}_{1\sigma}\cdot\nabla_{\bf
k}f({\bf k}))D({\bf r}_{21},\omega_{1},{\bf v}_{\bf
k})\hat{\varepsilon}_{2+}\cdot\nabla_{\bf
k}\theta(\hat{\varepsilon}_{2+}\cdot{\bf v}_{\bf k})+(1\leftrightarrow 2)$
(26)

where ${\bf r}_{21}=z_{2+}\hat{n}_{2+}-z_{1\sigma}\hat{n}_{1\sigma}$. To do
the integral over $z_{2+}$, note that given $z_{1\sigma}$, $z_{2+}$ is fixed
by $\delta(\hat{\bf v}_{\bf k}\times{\bf r}_{21})$. Using $\hat{\bf v}_{\bf
k}\times\hat{n}_{p\sigma}=-\sigma\hat{\bf v}_{\bf
k}\cdot\hat{\varepsilon}_{p\sigma}$, this is $\delta(-z_{2+}\hat{\bf v}_{\bf
k}\cdot\hat{\varepsilon}_{2+}+z_{1\sigma}\sigma\hat{\bf v}_{\bf
k}\cdot\hat{\varepsilon}_{1\sigma})$, so that
$z_{2+}=z_{1\sigma}\sigma\hat{\bf v}_{\bf
k}\cdot\hat{\varepsilon}_{1\sigma}/\hat{\bf v}_{\bf
k}\cdot\hat{\varepsilon}_{2+}$. It can then be checked that $|{\bf
r}_{21}|=\hat{\bf v}\cdot{\bf
r}_{21}=z_{1\sigma}\sigma\hat{\varepsilon}_{1\sigma}\times\hat{\varepsilon}_{2+}/\hat{\bf
v}_{\bf k}\cdot\varepsilon_{2+}$. Since $z_{1\sigma}$, $z_{2+}$ and $\hat{\bf
v}\cdot{\bf r}_{21}$ must all be positive, we require ${\rm
sgn}(\hat{\varepsilon}_{1\sigma}\times\hat{\varepsilon}_{2+})={\rm
sgn}(\hat{\bf v}_{\bf k}\cdot\hat{\varepsilon}_{1\sigma})=\sigma{\rm
sgn}(\hat{\bf v}_{\bf k}\cdot\hat{\varepsilon}_{2+})$. We then have

$\displaystyle\alpha(\left\\{\omega_{p}\right\\})=$
$\displaystyle\frac{e^{3}/\hbar^{2}}{2\eta-i(\omega_{1}+\omega_{2})}\sum_{{\bf
k},\sigma}\Theta_{\sigma}({\bf v}_{\bf
k})\int_{0}^{\infty}dz_{1\sigma}(\hat{\varepsilon}_{1\sigma}\cdot\nabla_{\bf
k}f({\bf
k}))\frac{e^{(i\omega_{1}-\eta)z_{1\sigma}\frac{\sigma\hat{\varepsilon}_{1\sigma}\times\hat{\varepsilon}_{2+}}{|{\bf
v}_{\bf k}|\hat{\bf v}\cdot\hat{\varepsilon}_{2+}}}}{|{\bf v}_{\bf
k}||\hat{\bf v}_{\bf
k}\cdot\hat{\varepsilon}_{2+}|}\hat{\varepsilon}_{2+}\cdot\nabla_{\bf
k}\theta(\hat{\varepsilon}_{2+}\cdot{\bf v}_{\bf k})$
$\displaystyle+(1\leftrightarrow 2)$ (27)

where $\Theta_{\sigma}({\bf v}_{\bf
k})=\theta((\hat{\varepsilon}_{1\sigma}\times\hat{\varepsilon}_{2+})(\hat{\bf
v}_{\bf
k}\cdot\hat{\varepsilon}_{1\sigma}))\theta(\sigma(\hat{\varepsilon}_{1\sigma}\times\hat{\varepsilon}_{2+})(\hat{\bf
v}_{\bf k}\cdot\hat{\varepsilon}_{2+}))$. We next use $\nabla_{\bf k}f({\bf
k})=\hat{\bf v}_{\bf k}(\hat{\bf v}_{\bf k}\cdot\nabla_{\bf k})f({\bf k})$,
along with the fact that $\nabla_{\bf k}\theta(\hat{\varepsilon}_{2+}\cdot{\bf
v}_{\bf k})$ will only be non-zero for $\hat{\bf v}_{\bf
k}\sim\pm\hat{n}_{2+}$, so that $\hat{\bf v}_{\bf
k}\cdot\hat{\varepsilon}_{1\sigma}\sim\pm\hat{\varepsilon}_{1\sigma}\times\hat{\varepsilon}_{2+}$
the integral over $z_{1\sigma}$ becomes

$\alpha(\left\\{\omega_{p}\right\\})=\frac{e^{3}/\hbar^{2}}{(2\eta-i(\omega_{1}+\omega_{2}))(\eta-i\omega_{1})}\sum_{{\bf
k},\sigma}M_{\sigma}\Theta_{\sigma}({\bf v}_{\bf
k})(\hat{n}_{2+}\cdot\nabla_{\bf k}f({\bf
k}))\hat{\varepsilon}_{2+}\cdot\nabla_{\bf
k}\theta(\hat{\varepsilon}_{2+}\cdot{\bf v}_{\bf k})+(1\leftrightarrow 2),$
(28)

where we have defined $M_{\sigma}={\rm
sgn}(\hat{\varepsilon}_{1\sigma}\times\hat{\varepsilon}_{2+})$, and now
$\Theta_{\sigma}({\bf v}_{\bf k})=\theta({\bf v}_{\bf
k}\cdot\hat{n}_{2+})\theta(\sigma M_{\sigma}{\bf v}_{\bf
k}\cdot\hat{\varepsilon}_{1\sigma})$. The sum over $\sigma=\pm 1$ in the first
term now depends on the angles $\varphi_{2}$ and $\varphi_{3}$. If both
$\varphi_{2,3}$ are less than $\pi$ (as shown in Fig. B1), then
$M_{\sigma}=1$, and $\sum_{\sigma}M_{\sigma}\theta(\sigma M_{\sigma}{\bf
v}_{\bf k}\cdot\hat{\varepsilon}_{2+})=1$. On the other hand, if
$\varphi_{2}>\pi$ ($\varphi_{3}>\pi$), then $M_{\sigma}=-\sigma$
($M_{\sigma}=+\sigma$). In this case $\sum_{\sigma}M_{\sigma}\theta(\sigma
M_{\sigma}{\bf v}_{\bf k}\cdot\hat{\varepsilon}_{2+})=0$. If we define
$\xi_{p}=\theta(\pi-\varphi_{p})$, then we conclude that

$\alpha(\left\\{\omega_{p}\right\\})=\frac{(e^{3}/\hbar^{2})\xi_{2}\xi_{3}}{(2\eta-i(\omega_{1}+\omega_{2}))(\eta-i\omega_{1})}\sum_{\bf
k}(\hat{n}_{2+}\cdot\nabla_{\bf k}f({\bf k}))\theta(\hat{\bf v}_{\bf
k}\cdot\hat{n}_{2+})\hat{\varepsilon}_{2+}\cdot\nabla_{\bf
k}\theta(\hat{\varepsilon}_{2+}\cdot{\bf v}_{\bf k})+(1\leftrightarrow 2)$
(29)

Following the analysis of Eq. 8, the sum on ${\bf k}$ in the numerator gives
$\chi_{F}/(2\pi)^{2}$, so we obtain

$\alpha(\left\\{\omega_{p}\right\\})=\frac{\chi_{F}e^{3}/h^{2}}{2\eta-i(\omega_{1}+\omega_{2})}\left[\frac{\xi_{2}\xi_{3}}{\eta-i\omega_{1}}+\frac{\xi_{1}\xi_{3}}{\eta-i\omega_{2}}\right]$
(30)

For the case where all angles are less than $\pi$ we recover Eq. 1,

$\alpha(\left\\{\omega_{p}\right\\})=\frac{\chi_{F}e^{3}/h^{2}}{(\eta-i\omega_{1})(\eta-i\omega_{2})}$
(31)

Note that for the special case treated in Eq. 6-10, with $\varphi_{1}=\pi$,
when the $V_{2}$ pulse follows the $V_{1}$ pulse, the excess charge $Q_{3}$
depends only on the first term in the brackets of (30), and is insensitive to
$\varphi_{1}=\pi$.