# A priori obstructions to resolution of singularities

Tobias Shin

###### Abstract

We present an argument due to Thom to formulate a priori cohomology
obstructions for a projective variety to admit an embedded resolution of
singularities, and generalize the argument to a field of characteristic $p>0$.
We show that these obstructions are defined for a general cohomology theory
that satisfies localization, étale excision, and $\mathbb{A}^{1}$-invariance,
and is equipped with a proper pushforward. As examples, we show that odd
degree Steenrod homology operations on higher Chow groups mod $\ell$, defined
by a suitable $E_{\infty}$-algebra structure, vanish on the fundamental class
of a smooth variety, where $\ell$ is any prime. We extend this general
vanishing result to arbitrary varieties for operations defined mod $\ell\neq
p$. Along the way, we also obtain a Wu formula for the mod $p$ Steenrod
operations in the case of closed embeddings of smooth varieties.

###### Contents

  1. 1 Introduction
  2. 2 Motivic cohomology and proper pushforward
    1. 2.1 Higher Chow groups

##  1 Introduction

The aim of this paper was to naïvely define topological obstructions for a
variety to admit a resolution of singularities over an algebraically closed
field $k$ of characteristic $p>0$. We do not succeed in this endeavor;
instead, we present a vanishing theorem for generalized cohomology theories
satisfying suitable axioms. As an example, we can consider motivic cohomology
with coefficients mod $\ell$ where $\ell$ is any prime (including $\ell=p$).
There exist mod $\ell$ Steenrod operations $Q^{\bullet}$ on motivic
cohomology, arising from an $E_{\infty}$-algebra structure (see Theorem
LABEL:SteenrodOps). We let $\beta$ denote the Bockstein and its dual.

######  Theorem 1.1.

Let $X\hookrightarrow P$ be a closed embedding of a variety $X$ into a smooth
variety $P$ over an algebraically closed field of characteristic $p>0$. Let
$Q^{\bullet}$ be the simplicial Steenrod operations on motivic cohomology
$H^{*}_{X}(P,\mathbb{Z}/\ell(*))$, where $\ell\neq p$, and let $Q_{\bullet}$
denote the dual operations on $H_{*}(X,\mathbb{Z}/\ell(*))$. The odd degree
operations $\beta Q_{s}$ vanish on the fundamental class $[X]$. If $X$ is also
assumed smooth, then the odd degree operations $\beta Q_{s}$ vanish on the
fundamental class $[X]$ for $\ell=p$.

Theorem 1.1 is a special case of Theorem 1.2, which applies to generalized
cohomology theories in the sense of Panin–Smirnov (see Definition
LABEL:generalizedcohomologytheory). Here, a cohomology theory $A$ is a
contravariant functor that has a long exact sequence of a pair, and satisfies
étale excision and $\mathbb{A}^{1}$-invariance. Examples of such theories can
be found in the article of Panin [panin2003oriented], such as classical
singular cohomology over $\mathbb{C}$, mod $\ell\neq p$ étale cohomology over
a field of characteristic $p>0$, higher Chow groups with integral coefficients
or coefficients mod $\ell$ where $\ell$ is any prime (including when
$\ell=p$), algebraic $K$-theory, algebraic cobordism theory, any representable
theory, etc. We further want our cohomology theory to have a ring structure
and to come from the cohomology of a sheaf (or hypercohomology of a complex of
sheaves) (see Definitions LABEL:ringcohomologydefinition and
LABEL:sheafcohomologydefinition). Moreover we require our theory $A$ to be
equipped with a proper pushforward (Definition
LABEL:generalizedproperpushforwarddefinition). It is then a theorem of
Panin–Smirnov that this is equivalent to equipping $A$ with a notion of Chern
classes, which is also equivalent to equipping $A$ with a notion of Thom class
and Thom isomorphism (see Theorem LABEL:paninSmirnovMainTheorem).

######  Theorem 1.2.

Let $A$ be a bigraded ring cohomology theory in the sense of Panin–Smirnov,
that is given by a sheaf cohomology theory valued in modules over a field, and
is equipped with a proper pushforward. Suppose $A^{i,j}(\textnormal{Spec
}k)=0$ for $i\neq 0$. Let $\phi=\phi^{\bullet}$ be a ring operation with well-
defined Todd genus that preserves even degrees in the first index, and let
$\beta$ be a natural operation that sends bidegrees $(i,j)$ to $(i+1,j)$ that
commutes with proper pushforward. Moreover, assume $\beta$ is a derivation
that vanishes on the Todd class of any bundle. Suppose $X\hookrightarrow P$ is
a closed embedding of a smooth variety $X$ into a smooth variety $P$. The
operation $\beta\phi^{s}$ vanishes on the dual class $\tau\in A^{2c,c}_{X}(P)$
where $c$ is the codimension of $X$ in $P$.

######  Theorem 1.3.

Let $A$ be a theory with the same hypotheses as in Theorem 1.2. Suppose
$X\hookrightarrow P$ is a closed embedding of a (possibly singular) variety
$X$ into a smooth variety $P$ over a field of characteristic $p>0$. If $A$ is
valued in $\mathbb{Z}/\ell$-modules, where $\ell\neq p$, then the operation
$\beta\tilde{\phi}^{s}$ vanishes on the dual class $\tau\in A^{2c,c}_{X}(P)$,
where $\tilde{\phi}$ is the operation associated to $\phi$ that commutes with
proper pushforward. Moreover, if $A$ is valued in $\mathbb{Z}/p$-modules and
$X$ admits an embedded resolution of singularities, then the operation
$\beta\tilde{\phi}^{s}$ vanishes on the dual class $\tau\in A^{2c,c}_{X}(P)$.

Theorem 1.3 essentially defines a priori cohomological obstructions to a
projective variety $X$ admitting an embedded resolution of singularities. If
$X$ admits a resolution, then certain mod $p$ cohomological operations must
vanish. Thus, in theory, one could construct an example of a singular variety
$X$ where these operations are non-zero. As of this writing, and to the
author’s knowledge, the problem of resolution of singularities in positive
characteristic is still open; if proven true, then Theorems 1.1, 1.2, and 1.3
can be interpreted as vanishing theorems.

However, the only examples we have either already vanish for reasons other
than resolution of singularities, or do not satisfy the hypotheses of the
theorems: the mod $\ell$ operations on motivic and étale cohomology vanish due
to alterations, and we do not know how to pushforward the mod $p$ operations.

Our motivation begins with the work of Thom [thom1952espaces]. In private
communication, Sullivan informed the author of his belief that Thom, after his
work on Steenrod’s problem of representing homology classes by manifolds, was
motivated to formulate cohomological obstructions to resolving the
singularities of a compact complex algebraic variety; Sullivan explains this
belief explicitly in the article [sullivan2004rene]. The cohomology
obstructions are defined as follows: first, for a given compact complex
algebraic variety $X$, one topologically embeds $X$ into a large euclidean
space $\mathbb{R}^{N}$. There are mod $\ell$ Steenrod operations $Q^{\bullet}$
defined on the cohomology of a pair $(\mathbb{R}^{N},\mathbb{R}^{N}-X)$, for
any prime $\ell$. By Alexander–Lefschetz–Poincaré duality, these cohomology
operations dualize to mod $\ell$ operations $Q_{\bullet}$ on the (Borel–Moore)
homology of $X$. These operations do not depend on the embedding or dimension
of euclidean space. Moreover, these homology operations commute with proper
pushforwards. Sullivan then explained the following argument, which he
attributed to Thom. We abuse notation and let $\beta$ denote the dualized
Bockstein operation on homology.

######  Proposition 1.1.

Let $X$ be a compact complex algebraic variety. Then the operations of odd
degree $\beta Q_{\bullet}$ vanish on the fundamental class $[X]$.

###### Proof.

We first prove the case when $X$ is smooth. We smoothly embed $X$ into a large
euclidean space $\mathbb{R}^{n}$. Since $X$ is complex, its normal bundle is
stably almost complex, so $N_{X}\mathbb{R}^{n}\oplus\mathbb{R}^{k}$ is a
complex bundle for some $k$. Embed $X$ into the larger euclidean space
$\mathbb{R}^{N}$ where $N=n+k$, so that its normal bundle is complex. By
definition of the operations, it suffices to show $\beta Q^{\bullet}$ vanishes
on the dual class $\tau$ in the relative cohomology
$H^{*}(\mathbb{R}^{N},\mathbb{R}^{N}-X,\mathbb{Z}/\ell)$. We have a natural
isomorphism $H^{*}(\mathbb{R}^{N},\mathbb{R}^{N}-X,\mathbb{Z}/\ell)\cong
H^{*}(N_{X}\mathbb{R}^{N},N_{X}\mathbb{R}^{N}-0,\mathbb{Z}/\ell)$ where
$N_{X}\mathbb{R}^{N}$ denotes the normal bundle of $X$ with respect to our
embedding. In this equivalence, the dual class $\tau$ is identified with the
Thom class of the normal bundle, so we will use the same notation for both.

The normal bundle $N_{X}\mathbb{R}^{N}$ induces a classifying map
$X\rightarrow Gr(c)$ where $Gr(c)$ is the infinite Grassmannian of complex
$c$-dimensional subspaces, where $c$ is the complex codimension of $X$ in
$\mathbb{R}^{N}$.

${N_{X}P}$${\gamma^{c}}$${X}$${Gr(c)}$

where $\gamma^{c}$ is the tautological bundle of complex $c$-dimensional
subspaces. The classifying map induces a map on the Thom spaces, and so a map
$H^{*}(\gamma^{c},\gamma^{c}-0,\mathbb{Z}/\ell)\rightarrow
H^{*}(N_{X}\mathbb{R}^{N},N_{X}\mathbb{R}^{N}-0,\mathbb{Z}/\ell)$, and sends a
cohomology class $\gamma$ in cohomology of the the universal Thom space to the
Thom class $\tau$ of the normal bundle, by functoriality of the Thom
isomorphism. The cohomology of the complex Grassmannian $Gr(c)$ is
concentrated in even degrees, and so operations $\beta Q^{\bullet}$ vanish on
$\gamma$. By naturality, they then vanish on the Thom class $\tau$, and so the
dual homology operations $\beta Q_{\bullet}$ vanish on the fundamental class
$[X]$.

Suppose now $X$ is singular. Then $X$ has a resolution of singularities $Y$ by
Hironaka [hironaka1964resolution], where $Y$ is smooth and the map
$Y\rightarrow X$ is proper and birational. In particular, the fundamental
class $[Y]$ is sent to $[X]$. The odd degree operations vanish on $[Y]$ by the
argument above, and they commute with proper pushforward, so they also vanish
on $[X]$. ∎

If $X$ is a singular variety that admits a resolution, then the odd degree
homology operations must vanish on its fundamental class. Thus, these
operations are “obstructions” in the sense that, finding an example of a
variety where these operations are non-zero would give a variety that had no
resolution. They are “a priori” as they do not require resolution of
singularities to be defined.

The idea of this paper is to translate this differential topology argument
into algebraic geometry, over a field of characteristic $p>0$. The central
difficulty was, and still may be, finding the correct cohomology theory
suitable for this program. Initially, Sullivan considered using étale
cohomology [sullivan2004rene] but learned from de Jong [de1996smoothness] that
a priori mod $\ell\neq p$ obstructions would vanish. The argument then
required a suitable $p$-adic cohomology theory, which captured $p$-torsion
information. There is a large amount of literature in the field exploring the
tension between a “nice” $p$-adic cohomology theory, Steenrod operations, and
resolution of singularities (for example, see the articles
[haution2012reduced], [ertl2021integral], [abe2021integral], [niziol2006p]).
We believe this tension is related to the Thom argument. That is, given a
sufficiently robust $p$-adic cohomology theory, we believe one can formulate
Thom obstructions.

A suitable $p$-adic cohomology theory for this program should have something
similar to Steenrod operations, although any graded ring endomorphism should
do. This ruled out considering theories which used rational coefficients, such
as rigid cohomology or Monsky–Washnitzer cohomology. Motivated by the work of
Mandell [mandell2001algebras], we were led to look for $p$-adic theories that
came from $E_{\infty}$-algebra structures. As a result, Sullivan and the
author were led to prismatic cohomology [bhatt2022prisms], crystalline
cohomology, and general Weil cohomology theories, but found that naïve
topological constructions failed, as ideas such as excision or local
contractibility did not hold. For example, the affine line has
infinite–dimensional prismatic cohomology, which made local constructions
difficult.

After initial difficulties with prismatic cohomology, we were led to consider
logarithmic de Rham–Witt forms [illusie1979complexe], [kato1982symmetric] and
observed that naïve topological ideas worked after taking fixed points of the
(inverse) Cartier operator. To the author, this looked like taking fixed
points of Frobenius, which kept contributing infinite–dimensional cohomology
for affine varieties. Stripping away the Frobenius, the author was led to
observe that the affine line was acyclic with respect to mod $p$ log de
Rham–Witt cohomology, which is equivalent to mod $p$ motivic cohomology.
Unfortunately, most of the contemporary literature on motivic cohomology takes
a highly categorical point of view, with a substantial amount of results
depending on resolution of singularities. As a result, we have opted to
present the material we need as simple and geometric as possible, making sure
none of the results we used depend on resolution. For example, we take motivic
cohomology as the higher Chow groups of Bloch [bloch1986algebraic] and avoid
the $cdh$-topology at all costs. This is the reason for the entirely
expository Section 2.

Aside from finding the right cohomology theory for the project, the main
difficulties of translating the argument of Proposition 1.1 from differential
topology to algebraic geometry over characteristic $p>0$ were in determining
which topological arguments were still allowed. For example, in Thom’s
original argument, one defines the homology operations by embedding a variety
into a large euclidean space. This step is not possible in algebraic geometry
unless the variety is affine. However, for our purposes, it suffices to
considers pairs $(P,X)$ where $X$ is a variety with a closed embedding into a
smooth variety $P$. By considering pairs, we could try and formulate
obstructions to an embedded resolution of singularities. Moreover, Thom uses a
classifying map to a Grassmannian, which again does not exist in algebraic
geometry without passing to stacks; however, we bypass this difficulty in
section LABEL:deligneSullivanEmbeddingBundletrick with an observation due to
Deligne–Sullivan, as long as our cohomology is homotopy invariant.

Thus, one needs a homotopy invariant $p$-adic cohomology theory that has a
notion of cohomology for a pair, which fits into a long exact sequence.
Moreover, this theory only needs to be defined on the category of smooth
varieties, since we can handle the Borel–Moore homology of a singular variety
$X$ embedded in a smooth variety $P$ by working with the cohomology of a pair
$A(P,P-X)$. For $X$ embedded in a smooth variety $P$, there is a notion of
motivic cohomology with support, which is naturally dual to the higher Chow
groups of the subvariety $X$. This fits the picture of higher Chow groups
behaving more like a Borel–Moore homology theory than a cohomology theory;
thus we can avoid any complications of having to work with motivic cohomology
of a singular variety, which was not even well defined until recent work of
Elmanto and Morrow [elmanto2023motivic].

By work of Joshua [joshua2001motivic] and Kriz–May [kriz1995operads], there
are Steenrod operations $Q^{\bullet}$ on motivic cohomology mod $\ell$ for
smooth varieties, where $\ell$ is any prime (including the characteristic).
The operations are derived from the product structure on Bloch’s cycle
complexes, which is only well defined up to quasi-isomorphism. This gives the
cycle complexes an $E_{\infty}$-algebra structure, which then inherits
generalized Steenrod operations by work of May [may1970general]. We examine
these operations closely in Section LABEL:Steenrod and extend them to
cohomology with support. We emphasize here that these Steenrod operations are
not the same as those defined by Voevodsky [voevodsky2003reduced] or Primozic
[primozic2020motivic], as those operations vanish on fundamental classes for
trivial degree reasons (see Remark LABEL:vanishForTrivialReasons).

However, homotopy theory is one thing, but geometry is another. Once the
operations are defined for cohomology with support, there is now another
nuance; we thank Patrick Brosnan for informing us of this crucial point. In
classical topology, one can define the Borel–Moore homology groups of a space
$X$ as the Alexander–Lefschetz–Poincaré dual of the cohomology of a pair
$(\mathbb{R}^{N},\mathbb{R}^{N}-X)$. The groups are then proved to be
independent of the choice of embedding. The cohomology of a pair admits
Steenrod operations; however, the dualized homology operations do depend on
the embedding and ambient space. In particular, the operations do not commute
with proper pushforward. At best, there is a Riemann–Roch type formula that
describes their failure to commute with pushforward, with error terms arising
from characteristic classes of the tangent bundle of the ambient space. We
call these error terms a Todd genus (see Definition
LABEL:toddgenusdefinition). Thus, if one embeds into euclidean space, the
error terms all vanish. The work of Panin–Smirnov [panin2000oriented],
[panin2003oriented], [panin2004riemann], [panin2009oriented],
[smirnov2007riemann] gives a general Riemann–Roch type formula for generalized
cohomology theories in the sense above, given a notion of proper pushforward
on cohomology (see Definition LABEL:generalizedproperpushforwarddefinition)
and a suitable ring operation $\phi$ (see Definition
LABEL:ringcohomologydefinition). Given such a Riemann–Roch type formula, one
can twist the operation by the Todd genus, and we can define new operations
$\tilde{\phi}$ that do commute with pushforward. It is through these
Riemann–Roch type formulas that the geometry of the ambient space is crucial
to our homology operations. This is the content of Section
LABEL:RiemannRochTypeFormulasSection.

This paper is then about transporting the entire framework of differential
topology to algebraic geometry, to then mimic the argument of Thom in
Proposition 1.1. Moreover, once this is done, one can use alterations of de
Jong [de1996smoothness] with appropriate degree [temkin2017tame] to extend the
vanishing results to singular varieties, so long as the coefficients are
valued mod $\ell\neq p$. The proofs of the Theorems 1.1, 1.2, and 1.3 are the
content of Sections LABEL:embeddingBundlesSection and LABEL:ThomObsVanish.

As mentioned prior, we do not know how to pushforward the mod $p$ motivic
operations, as there is no Riemann–Roch type formula. This is discussed in the
latter parts of section LABEL:RiemannRochTypeFormulasSection. Although there
is no general Riemann–Roch type formula for the mod $p$ operations for general
proper pushforwards, we are able to obtain one for the special case of closed
embeddings of smooth varieties. This is proven as Theorem
LABEL:WuformulaforSimplicialSteenrodOps.

######  Theorem 1.4.

(Wu formula) Let $i:X\hookrightarrow P$ be a closed embedding of smooth
varieties, and let $i_{!}$ denote the proper pushforward on mod $p$ motivic
cohomology. We obtain the formula

$Q^{\bullet}i_{!}(\alpha)=i_{!}(Q^{\bullet}(\alpha)\cup
c_{\textnormal{top}}^{p-1}(N_{X}P))$

for $\alpha\in H^{*}(X,\mathbb{Z}/p(*))$, where $Q^{\bullet}$ are the mod $p$
simplicial operations on mod $p$ motivic cohomology, and
$c_{\textnormal{top}}(N_{X}P)$ is the top Chern class of the normal bundle of
$X$ in $P$.

Finally, we would like to say that our approach is naïve, in that we have
given a set of axioms sufficient but not necessary for Thom’s argument to
work. For example, one could relax the homotopy invariance axiom and still
obtain many of the formal results needed for the argument (see Section
LABEL:conjecturesandcounterexamples). Our approach is naïve in the sense that
having the homotopy invariance axiom allows for simpler arguments. It is still
possible for there to be a more general cohomology theory with a suitable
choice of ring operation that succeeds where we have failed. It is also
entirely true of course that the program of finding obstructions is vacuous,
if resolution of singularities holds true, in which case we simply have these
vanishing theorems.

We also want to note the similiarity in spirit of our work with that of Fulton
(Example 19.1.8 in [fulton2013intersection]), Kawai [kawai1977note], Benoist
[benoist2022steenrod], Quick [quick2011torsion], and Atiyah–Hirzebruch
[atiyah1961cohomologie] who all utilize Steenrod operations as obstructions to
cohomology classes being represented by subvarieties (for example, disproving
the integral Hodge conjecture).

Structure of the paper. Sections 2 and LABEL:Steenrod are entirely expository,
with nothing new in them. The expert in higher Chow groups and its formal
properties should feel free to skip section 2, and the expert in Steenrod
operations should feel free to skip the first half of Section LABEL:Steenrod.
The first half of Section LABEL:Steenrod concerns results that are known, but
either hard to find in the literature or in unpublished material. The real
meat of the paper is in the latter half of Section LABEL:Steenrod, starting
with section LABEL:AxiomaticApproach. The notion of generalized cohomology
theory is given in section LABEL:AxiomaticApproach, and the Riemann–Roch type
theorems of Panin–Smirnov are discussed in Section
LABEL:RiemannRochTypeFormulasSection. Section LABEL:embeddingBundlesSection
discusses an observation due to Deligne–Sullivan that allows us to use
classifying maps in algebraic geometry, and section LABEL:ThomObsVanish brings
all aforementioned sections together to generalize the Thom argument for
generalized cohomology theories. It also proves the Thom argument for certain
Steenrod operations defined on motivic cohomology. Section
LABEL:alterationsSection shows the mod $\ell\neq p$ operations all vanish.
Section LABEL:conjecturesandcounterexamples is purely conjectural and has no
meaningful content.

Notation. Throughout this paper, we fix an algebraically closed field $k$ of
characteristic $p>0$. A variety $X$ is an integral, separated scheme over $k$
of finite type. All varieties are quasiprojective unless specified otherwise.
All products and dimensions of spaces are over the field $k$. Likewise, all
tensor products of abelian sheaves are over $\mathbb{Z}$ or $\mathbb{Z}/\ell$,
where $\ell$ is any prime.

Acknowledgements. We would like to thank Dennis Sullivan, who originally
allowed the author to work on this project as his thesis, and for telling us
the Thom argument and the Deligne–Sullivan trick. We would like to also thank
Thomas Geisser and Marc Levine for patiently answering the author’s basic
questions on higher Chow groups, Patrick Brosnan for teaching us that the dual
Steenrod homology operations on Borel–Moore homology do not commute with
proper pushforward, Peter May for helping the author understand some nuances
about $E_{\infty}$-algebras, and Spencer Bloch for clarifying some points in
his private letter. We also want to thank Benson Farb for helpful comments on
an initial draft of this paper.

##  2 Motivic cohomology and proper pushforward

Most of the material in this mainly expository section can be found in work of
Bloch [bloch1986algebraic], Geisser [geisser2004motivic],
[geisser2010duality], Levine [levine1998mixed], [levine2001techniques], and
Mazza–Weibel–Voevodsky [mazza2006lecture]. We will discuss the basic
properties of higher Chow groups and motivic cohomology, such as pullback,
proper pushforward, cup products, projection formulas, and twisted Poincaré
duality in the sense of Bloch–Ogus [bloch1974gersten]. This twisted duality
will allow us to analyze singular varieties that admit embeddings into smooth
ones. We will also discuss how the two theories are related for smooth
varieties. For a nice summary of the properties of higher Chow groups, see
Part I, Chapter II.2 in Levine [levine1994bloch].

###  2.1 Higher Chow groups

We first recall the definition of Bloch’s cycle complex [bloch1986algebraic],
following Geisser [geisser2010duality] and Levine [levine1998mixed].

######  Definition 2.1.

The algebraic $n$-simplex is defined as the affine hyperplane
$\textnormal{Spec }k[t_{0},...,t_{n}]/(t_{0}+...+t_{n}=1)$. A face $F$ of
$\Delta^{n}$ is a subscheme defined by equations of the form
$t_{i_{0}}=...=t_{i_{r}}=0$ (or $\Delta^{n}$ itself). The vertices $v_{i}$ are
the faces given by $t_{j}=0$ for $j\neq i$. An order-preserving map of
simplices $g:[n]\rightarrow[m]$ defines a map
$\Delta^{n}\rightarrow\Delta^{m}$ by defining a map on the vertices
$v^{n}_{i}\rightarrow v^{m}_{g(i)}$ and then extending affine-linearly. We
write $\Delta^{\bullet}$ as the cosimplicial scheme
$[n]\rightarrow\Delta^{n}$.

######  Definition 2.2.

Let $X$ be a variety. Define the cycle complexes as follows: let $z_{n}(X,i)$
be the free abelian group generated by closed integral subschemes $W$ of
dimension $n+i$ in $X\times\Delta^{i}$ such that $\textnormal{dim}W\cap
X\times F\leq n+i+\textnormal{dim}X\times F-\textnormal{dim}X\times\Delta^{i}$
for all faces $F$ of $\Delta^{i}$, i.e., $W$ meets all faces properly in
$X\times\Delta^{i}$.

We obtain a chain complex $z_{n}(X,*)$ by taking as our differential the
alternating sum of intersections with faces, which we call the $n$-th cycle
complex. That is, $\partial(W)=\sum_{j}[W\cap V(t_{j})]$, where $V(t_{j})$ is
the variety determined by $t_{j}=0$, and where $[W\cap V(t_{j})]$ is the
linear combination of the reduced irreducible components of the intersection
$W\cap V(t_{j})$, with coefficients given by intersection multiplicities. The
higher Chow groups are defined as the homology of this chain complex, with
$CH_{n}(X,i)$ denoting the $i$-th homology group. We call this the $i$-th
higher Chow group of $n$-cycles.

######  Remark 2.1.

We can also grade the cycle complex by codimension, with $z^{n}(X,i)$ denoting
the free abelian group generated by closed integral subschemes $W$ of
codimension $n$ in $X\times\Delta^{i}$ such that $W\cap X\times F$ has
codimension $\geq n$ in $X\times F$ for all faces $F$, i.e., $W$ meets all
faces $X\times F\subset X\times\Delta^{i}$ properly. This again yields a
homological chain complex $z^{n}(X,*)$, whose $i$-th homology we write as
$CH^{n}(X,i)$. By definition, we have that $z_{n}(X,i)=z^{d-n}(X,i)$ for a
pure $d$-dimensional variety $X$. Note in this case we have an equality on the
underlying sets, and that the regrading is purely formal. We will see later
that Poincaré duality for smooth varieties recovers this abstract isomorphism
of groups through a suitable pairing defined by a product structure; this
product structure is not necessarily available for singular varieties,
although the abstract isomorphism of groups still is.

Taking the $0$-th homology for the cycle complexes $z_{n}(X,*)$ imposes the
relation of rational equivalence on subschemes of $X$, thereby recovering the
ordinary Chow groups $CH_{n}(X,0)\cong CH_{n}(X)$.

######  Proposition 2.1.

The cycle complexes and homology groups satisfy the following properties:

  1. 1.

(Proper pushforward) If $f:X\rightarrow Y$ is a proper map, then the map
$f\times Id:X\times\Delta^{i}\rightarrow Y\times\Delta^{i}$ is proper. This
induces a pushforward map on algebraic cycles $(f\times
Id)_{*}:z_{n}(X,i)\rightarrow z_{n}(Y,i)$ given by sending a closed subscheme
$W$ to $[k(W):k((f\times Id)(W))](f\times id)(W)$ if the dimension of
$(f\times Id)(W)$ equals the dimension of $W$, and $0$ otherwise. The map
$(f\times Id)_{*}$ commutes with the differentials (see Proposition 1.3 in
[bloch1986algebraic]). We abuse notation and denote the induced map on
homology by $f_{*}:CH_{n}(X,i)\rightarrow CH_{n}(Y,i)$, for every $n$ and
every $i$.

  2. 2.

(Flat pullback) If $f:X\rightarrow Y$ is a flat map (of relative dimension
$c$), then the map $f\times Id:X\times\Delta^{i}\rightarrow Y\times\Delta^{i}$
is flat. This induces a pullback map $(f\times Id)^{*}:z_{n}(Y,i)\rightarrow
z_{n+c}(X,i)$ given by sending a closed subscheme $W$ to $(f\times
Id)^{-1}(W)$. The map $(f\times Id)^{*}$ commutes with the differentials (see
Proposition 1.3 in [bloch1986algebraic]). We abuse notation and denote the
induced map on homology by $f^{*}:CH_{n}(Y,i)\rightarrow CH_{n+c}(X,i)$ for
every $n$ and every $i$.

  3. 3.

(Localization) For a closed immersion $X\xhookrightarrow{i}P$, the left exact
sequence

$0\rightarrow
z_{n}(X,*)\xrightarrow{i_{*}}z_{n}(P,*)\xrightarrow{j^{*}}z_{n}(P-X,*)$

defines a distinguished triangle in the derived category of abelian groups for
every $n$, where $i_{*}$ is the pushforward induced by the closed immersion,
and $j^{*}$ is the pullback induced by the open immersion
$(P-X)\xhookrightarrow{j}P$.

  4. 4.

(Homotopy) If $\pi:E\rightarrow X$ is a vector bundle over a smooth $X$, the
induced pullback $\pi^{*}$ on the cycle complexes is a quasi-isomorphism for
all $n$. In particular, (beware the grading!)
$CH^{n}(X,i)\xrightarrow{\pi^{*}}CH^{n}(E,i)$ is an isomorphism for all $i$
and $n$.


training with biased unlabeled data is highly risky since it may potentially
amplify the biased models predictions. This work explores how to mitigate the
effect of dataset bias in unlabeled data, and can potentially promote fair
self-training systems.


# Towards Detecting Need for Empathetic Response in
Motivational Interviewing

Zixiu Wu<EMAIL_ADDRESS>Philips ResearchEindhovenNetherlands University
of CagliariCagliariItaly ,  Rim Helaoui<EMAIL_ADDRESS>Philips
ResearchEindhovenNetherlands ,  Vivek Kumar<EMAIL_ADDRESS>,  Diego
Reforgiato Recupero<EMAIL_ADDRESS>and  Daniele Riboni
<EMAIL_ADDRESS>University of CagliariCagliariItaly

(2020)

###### Abstract.

Empathetic response from the therapist is key to the success of clinical
psychotherapy, especially motivational interviewing. Previous work on
computational modelling of empathy in motivational interviewing has focused on
offline, session-level assessment of therapist empathy, where empathy captures
all efforts that the therapist makes to understand the client’s perspective
and convey that understanding to the client. In this position paper, we
propose a novel task of turn-level detection of client need for empathy.
Concretely, we propose to leverage pre-trained language models and empathy-
related general conversation corpora in a unique labeller-detector framework,
where the labeller automatically annotates a motivational interviewing
conversation corpus with empathy labels to train the detector that determines
the need for therapist empathy. We also lay out our strategies of extending
the detector with additional-input and multi-task setups to improve its
detection and explainability.

motivational interviewing, empathy, classification, deep learning

††journalyear: 2020††copyright: acmcopyright††conference: Companion
Publication of the 2020 International Conference on Multimodal Interaction;
October 25–29, 2020; Virtual event, Netherlands††booktitle: Companion
Publication of the 2020 International Conference on Multimodal Interaction
(ICMI ’20 Companion), October 25–29, 2020, Virtual event, Netherlands††price:
15.00††doi: 10.1145/3395035.3425228††isbn: 978-1-4503-8002-7/20/10††ccs:
Computing methodologies Discourse, dialogue and pragmatics††ccs: Computing
methodologies Transfer learning††ccs: Applied computing Psychology

##  1\. Introduction

Empathy from the counsellor side is widely recognised as essential to building
counsellor-client rapport in psychotherapy (Elliott et al.,
2011)111https://www.stephenrollnick.com/three-pieces-on-empathy/. Its
importance is particularly acknowledged in motivational interviewing (Miller
and Rollnick, 2012) (MI), a psychotherapeutic technique that has proved
successful in helping people achieve positive behaviour change by eliciting
their own motivation. It is also a key aspect of psychotherapeutic interview
quality. For example, according to the most used Motivational Interviewing
Coding system (MITI (Moyers et al., 2016)), assessing MI integrity goes
through four dimensions, namely cultivating change talk, softening sustain
talk, partnership, and empathy222https://casaa.unm.edu/download/miti4_2.pdf.

While such assessments are mostly done by experts manually, recent research
has explored automatic analysis and rating of therapist empathy in MI based on
text (Xiao et al., 2012; Gibson et al., 2015, 2016), speech (Xiao et al.,
2014, 2015), or both (Xiao et al., 2016b). Nevertheless, these studies are
limited since 1) they only evaluate offline, session-level counsellor empathy,
instead of offering real-time advice; 2) they nearly all rely on classical
machine learning methods with heavy feature engineering, yet pre-trained
language models such as BERT (Devlin et al., 2019) have not been explored
despite their superior results in Natural Language Understanding tasks; 3)
they were conducted on undisclosed datasets of MI conversations, which makes
their results difficult to verify or replicate.

To address the limitations above, we introduce a novel text-based task of
detecting the need for empathy in immediate, turn-level MI counsellor
response, and plan to tackle it by fine-tuning large-scale pre-trained
language models on publicly available free-access datasets of general and
therapeutic dialogues. This new task can be highly valuable for educating
inexperienced coaches and providing them with real-time guidance in their
first actual sessions.

We will approach the task based on two recent empathy-related dialogue
datasets: PEC (Zhong et al., 2020) (Persona-based Empathetic Conversation) and
HighLowRolePlayMI (Pérez-Rosas et al., 2019). The former comprises general
conversations from Reddit333Reddit (https://www.reddit.com/) is an online
platform consisting of a number of subforums, a.k.a. subreddits, each
corresponding to a specific topic for Reddit users to discuss. with each
annotated as empathetic or non-empathetic at dialogue-level but not turn-
level. The latter consists of role-play demonstrations of high- and low-
quality MI counselling from YouTube, where each conversation (transcribed) has
only session-level “high” and “low” quality labels. We make the following
assumptions about these datasets:

  1. (1)

In the high-quality conversations in HighLowRolePlayMI (aliased as
HighRolePlayMI), the therapist only shows empathy when the client needs it.

  2. (2)

Responses from the listener are predominantly empathetic in the empathetic PEC
conversations and non-empathetic in the non-empathetic ones.

  3. (3)

How the skilled therapist in HighRolePlayMI decides whether to show empathy to
the client based on the conversation flow is different from the manner in
which the ordinary Reddit listener in the empathetic PEC conversations makes
this decision before reacting to the speaker.

  4. (4)

How the therapist shows empathy HighRolePlayMI is similar to the way the
listener shows empathy in the empathetic PEC conversations, meaning that the
empathetic responses in two different domains (therapeutic and general
conversations) share similar patterns.

Assumption (1) is highly reliable because of the importance of therapist
empathy in high-quality MI counselling. Assumption (2) is verified
statistically by Zhong et al. (2020) with “substantial” inter-annotator
agreement. Assumption (3) is also very likely to be true since skilled
therapists have been specially trained to engage their clients with empathy,
whereas ordinary Reddit users have not. Assumption (4) is reasonable but yet
to be substantiated, and we therefore will explore its reliability in our
experiments. Enabled by these assumptions, our approach consists of two steps:
empathy labelling of HighRolePlayMI exploiting PEC and need-for-empathy
detection using HighRolePlayMI.

For empathy labelling, we will annotate each therapist utterance in
HighRolePlayMI with a binary empathy-or-not label. Specifically, we will train
a binary classifier (labeller) to distinguish between responses from the
empathetic conversations and those from the non-empathetic ones in PEC, so
that the resultant model is able to differentiate between empathetic and non-
empathetic responses, based on assumption (2). The labeller will then label
each therapist response in HighRolePlayMI as empathetic or non-empathetic,
based on assumption (4). Note that due to assumptions (1) and especially (3),
we cannot directly train a detector on the empathetic dialogues of PEC
(general conversation domain) to accomplish the need-for-empathy detection
task on HighRolePlayMI (therapeutic dialogue domain).

If the performance evaluation of the labeller on HighRolePlayMI reaches
satisfactory levels (e.g. w.r.t. accuracy/F1), we will move on to the proposed
need-for-empathy detection task. More concretely, we will train a detector on
HighRolePlayMI which is now annotated with an empathy label for each therapist
utterance. Given the conversation history where the last turn is from the
client, the detector will reach a yes/no answer on whether empathy is needed
in the immediate therapist response to the latest client utterance.

Both the labeller and the detector will be based on pre-trained language
models and trained end-to-end. We will also experiment with additional-input
regimes and multi-task designs to enhance the ability of the detector and
increase its explainability.

In summary, our contributions are as follows:

  * •

We propose a new need-for-empathy detection task, which aims to provide the
human therapist with real-time advice on whether to show empathy in their
response to the client.

  * •

We propose an automatic empathy labelling method that could annotate therapist
empathy, using publicly available free-access datasets of empathy-related
general dialogues.

  * •

We present our plan for approaching the detection task, leveraging the
aforementioned automatic annotation and simply requiring free-access MI
dialogues in the public domain without manually created empathy labels.

  * •

We detail our proposed additional-input and multi-task detector extensions for
explainability improvement.

##  2\. Related Work

The work we propose is closely related to prior work on 1) data-driven
analysis of therapeutic empathy in MI and 2) text-based approaches to
computational empathy in general conversation, as we will address the former
using techniques inspired by the latter.

###  2.1. Data-Driven Approaches to Therapeutic Empathy Analysis for MI

Empathy is fundamental in coaching in general and in clinical counselling in
particular, such as evidence-based behavioural treatments like MI. Prior work
studying the link between empathy and MI delivery has focused on the speech
and language of the therapist.

An early text-based attempt (Xiao et al., 2012) adopted an n-gram language
model for binary classification of utterance empathy. Gibson et al. (2015)
used generalised linear regression on other linguistic features as well as
psycholinguistic norm features. More recently, long short-term memory networks
(LSTMs) (Hochreiter and Schmidhuber, 1997) were used for turn-level encoding
to infer utterance-level behaviours that are further used by a deep neural
network (DNN) to predict session-level empathy (Gibson et al., 2016).

The other line of work uses speech features. Xiao et al. (2014) extracted
prosodic features such as pitch and energy from speech signals for empathy
classification using linear Support Vector Machines. Speech rate entrainment
during dyadic interactions has also been investigated for its relationship to
therapist empathy (Xiao et al., 2015). Finally, Pérez-Rosas et al. (2017)
trained a random forest on both linguistic and acoustic features to identify
therapist empathy.

Note that previous work, as listed above, has all targeted offline therapist
empathy assessment or identification of contributing factors to high-quality
therapy. Our goal, in contrast, is to detect the need for therapist empathy,
which makes our proposal closer to the work of Cao et al. (2019), where a
forecaster of MI behavioural codes was developed to provide real-time guidance
for the therapist on what action (e.g. reflection/question/$\cdots$) to take
next. Our work differs from it in that we take an empathy-centred perspective.

It is worth mentioning that there is more recent work on deep-learning-powered
MI analysis, including text-based (Cao et al., 2019; Tanana et al., 2016; Xiao
et al., 2016a; Gibson et al., 2018), speech-based (Singla et al., 2020), and
multimodal (Chen et al., 2019) coding of therapist actions according to MI
behaviour codes, but we do not consider this sphere of research since it is
still predominantly for offline assessment of therapy quality and does not
involve explicit empathy modelling.

###  2.2. Text-Based Approaches to Computational Empathy in General
Conversation

Substantial progress has been achieved in text-based sentiment analysis (SA)
(Dridi and Reforgiato Recupero, 2019; Recupero et al., 2015; Recupero et al.,
2014; Reforgiato Recupero and Cambria, 2014) in recent years, exemplified by
the over-97% accuracies from the top entries in the SST-2 SA task on the GLUE
(Wang et al., 2019) leaderboard444https://gluebenchmark.com/leaderboard. In
comparison, text-based empathy analysis, especially for general conversation,
attracted less attention, until EmpatheticDialogues (Rashkin et al., 2019), a
dataset of empathetic conversations grounded in emotions and situations, was
made public. Since then, various studies have explored creating an empathetic
open-domain conversation agent. Therefore, we focus on the approaches to
computational empathy for such agents in this section.

Many of those agents incorporate current user emotion during their response
generation. An early study (Zhou and Wang, 2018) used emotion labels (emojis
in tweets) as extra input to train a conditional variational encoder for
response generation. Similarly, Lubis et al. (2018) created an user-emotion
context vector in addition to the HRED (Sordoni et al., 2015) response-
generation framework. In (Rashkin et al., 2019), the emotion of the user is
categorised as an emotion label to prepend each user utterance so that the
response generator attends to user emotion explicitly. With GPT (Radford et
al., 2018) as its backbone, CAiRE (Lin et al., 2020) adds an user-emotion-
detection auxiliary objective in addition to the conventional response
language modelling. Lin et al. (2019), on the other hand, lined up specialised
response generators that are each trained to reply to user utterances of a
unique emotion.

Desirable or future emotion of the user or sentiment-aware agent has also been
explored. Zandie and Mahoor (2020) encouraged the agent to learn an
appropriate emotion for its response, whereas Li et al. (2020) conditioned
their chatbot utterance on the desirable user emotion that the agent is trying
to elicit. Within a reinforcement learning framework, Shin et al. (2020)
rewarded response candidates likely to induce positive user emotion.

##  3\. Data

The datasets we have considered for our problem are introduced below. Among
them, PEC will be used for the labelling step and HighRolePlayMI for the
detecting step.

Persona-based Empathetic Conversation (PEC) (Zhong et al., 2020) features 291k
one-to-one general conversations crawled from 2 subreddits:
r/happy555https://www.reddit.com/r/happy/ and
r/offmychest666https://www.reddit.com/r/offmychest/, both considered to
consist mainly of empathetic conversations, as well as another 725k one-to-one
casual conversations from
r/CasualConversation777https://www.reddit.com/r/CasualConversation used as the
control group, based on the assumption that casual conversations are less
empathetic. The author verified the empathy and non-empathy of these
conversation corpora with the “substantial agreement” shown by Fleiss’ kappa
(Fleiss, 1971) among the empathy annotators. We further group the 291K
empathetic conversations as EmpPEC and the 725K non-empathetic ones as
NonEmpPEC.

Pérez-Rosas et al. (2019) created the first and, to the best of our knowledge,
only publicly available dataset of MI conversations, which we term
HighLowRolePlayMI. The dataset comprises 259 role-play conversations collected
from public video-sharing sources such as YouTube and Vimeo, with 155 labelled
as high-quality counselling and the other 104 as low-quality. The
conversations are available as both videos and transcripts, without any extra
annotation. We denote the high-quality counselling subset as HighRolePlayMI
and the low-quality subset as LowRolePlayMI.

##  4\. Proposed Methodology

Our method follows a labelling-detecting two-step setup. Both the labeller and
the detector will have pre-trained language models as their underlying
architectures. In particular, we will explore extending the detector with
additional-input regimes and multi-task designs that can boost its performance
and improve its explainability.

###  4.1. Step 1: Labelling

####  4.1.1. Task Definition

The labeller, $cls^{label}_{emp}$, will be trained to determine whether a
listener response is empathetic given the most recent speaker-listener talk
turns. More concretely, $cls^{label}_{emp}$ will be given as input
$H_{t,N}\allowbreak=\allowbreak\\{u^{S}_{t-N+1},\allowbreak
u^{L}_{t-N+1},\allowbreak\cdots,\allowbreak u^{S}_{t},\allowbreak
u^{L}_{t}\\}$, a small window of the latest $N$ (e.g. $N=2$) exchanges between
the interlocutors at time step $t$ where $u^{S}_{i}$ is the $i$-th speaker
utterance and $u^{L}_{j}$ is the $j$-th listener utterance, and produce a
binary classification on whether $u^{L}_{t}$ is an empathetic response to
$u^{S}_{t}$. Note that we provide the $2N$-utterance window to allow for more
context for the labeller.

####  4.1.2. Training

Using PEC to train $cls^{label}_{emp}$, we consider the initiator of a
conversation as the speaker and the other interlocutor as the listener. Every
listener response and its context in EmpPEC will be deemed a positive
(empathetic) example, and likewise each listener reply with its context in
NonEmpPEC will be used as a negative (non-empathetic) example, as shown in
Figure 1.

Figure 1. Training the labeller with a 2-utterance window over each PEC
conversation. Every listener reply in subreddits r/happy and r/offmychest is
considered an empathetic example, while every response in r/CasualConversation
is deemed non-empathetic.

We will use a pre-trained language model such as BERT (Devlin et al., 2019) or
GPT-2 (Radford et al., 2019) as the backbone of $cls^{label}_{emp}$. As input
to BERT, for example, we will represent a 4-turn conversation, assuming a
window length of 4, as $\\{[CLS],\allowbreak[S],\allowbreak
u^{S}_{1},\allowbreak[L],\allowbreak u^{L}_{1},[S],\allowbreak
u^{S}_{2},\allowbreak[L],\allowbreak u^{L}_{2}\\}$, where $[S]$, $[L]$, and
$[CLS]$ are special tokens. Specifically, $[S]$ indicates the utterance that
follows is from the speaker, the same goes for $[L]$ for the listener, while
the top-layer BERT representation of $[CLS]$888$[CLS]$ is a special token
required by BERT (Devlin et al., 2019) and its variants. It is placed at the
beginning of the input, and its top-layer BERT representation is usually seen
as a high-dimensional representation of the entire input. See (Devlin et al.,
2019) for more details. will be fed to a multi-layer perceptron (MLP) that
will ultimately decide if $u^{L}_{2}$ is an empathetic response to
$u^{S}_{2}$.

Once $cls^{label}_{emp}$ is trained, it will be used to annotate the MI
corpus, considering the client as the speaker and the therapist as the
listener, such that each therapist response in HighLowRolePlayMI is labelled
as empathetic or non-empathetic.

###  4.2. Step 2: Detection

####  4.2.1. Task Definition

Once the accuracy for Step 1 has been validated, we define the need-for-
empathy detection task as equivalent to determining whether the therapist
should show empathy in their immediate response to the client, given the
history of the conversations so far. More formally, we represent the
therapeutic dialogue history as
$H^{thr}_{t}\allowbreak=\\{u^{C}_{1},\allowbreak u^{T}_{1},\allowbreak
u^{C}_{2},\allowbreak u^{T}_{2},\allowbreak\cdots,\allowbreak
u^{C}_{t-1},\allowbreak u^{T}_{t-1},\allowbreak u^{C}_{t}\\}$, where
$u^{C}_{i}$ is the $i$-th client utterance and $u^{T}_{j}$ is the $j$-th
therapist utterance, and the task is to decide whether $u^{T}_{t}$, the
response to $u^{C}_{t}$, should be empathetic given $H^{thr}_{t}$, as
illustrated in Figure 2.

Figure 2. Training the detector with (partial) MI conversation history. The
“non-empathetic” and “empathetic” labels of the therapist utterances are given
by the labeller, with the client as the speaker and the therapist as the
listener.

####  4.2.2. Baseline

We will train a baseline detector $cls^{dct}_{emp}$ on the now-annotated
HighRolePlayMI to approach the task, with the input adapted from the
representation described in Section 4.1.2. More formally, we will allow a much
larger window of length $2M+1$ ($M\gg N$) for the most recent $2M+1$
utterances: $H^{thr}_{t,M}\allowbreak=\allowbreak\\{u^{C}_{t-M},\allowbreak
u^{T}_{t-M},\allowbreak\cdots,\allowbreak u^{C}_{t-1},\allowbreak
u^{T}_{t-1},\allowbreak u^{C}_{t}\\}$, and the input to the base pretrained
language model, in the case of BERT (Devlin et al., 2019) or its variants
(e.g. ALBERT (Lan et al., 2020), RoBERTa (Liu et al., 2019)), will be
$\\{[CLS],\allowbreak[C],\allowbreak u^{C}_{t-M},\allowbreak[T],\allowbreak
u^{T}_{t-M},\allowbreak\cdots,\allowbreak[C],\allowbreak
u^{C}_{t-1},\allowbreak[T],\allowbreak u^{T}_{t-1},\allowbreak[C],\allowbreak
u^{C}_{t}\\}$, where $\allowbreak[C]$ is the special token for the client and
$[T]$ for the therapist. The MLP above the language model will then reach a
binary classification using the representation of the $[CLS]$, on whether
$u^{T}_{t}$ needs to be empathetic.

Considering the potential susceptibility of this system to overfitting due to
the small size of HighRolePlayMI, we propose 2 extensions to the baseline:
emotion recognition and therapist response generation, as auxiliary objectives
or for additional input. Their details are explained in the remainder of this
section.

####  4.2.3. Extension: Emotion Recognition

Acknowledging emotions is essential to empathetic communication (Rashkin et
al., 2019). Considering that emotion recognition (Recupero et al., 2019) is a
well-resourced task, we hypothesise that involving emotions explicitly will be
beneficial for the learning process of the detector and may enable more
explainability.

For additional input to the baseline, an external emotion classifier
$cls_{emt}$ will be trained on an emotion-labelled conversation dataset, e.g.
EmpatheticDialogues (Rashkin et al., 2019), and provide an emotion label
prepended at the beginning of each utterance before the conversation window is
fed to $cls^{dct}_{emp}$ as the input, similar to (Rashkin et al., 2019).

In a multi-task setting, we will feed the representation of $[CLS]$ to two
MLPs: one for need-for-empathy detection as in the baseline, and the other for
emotion recognition. By assigning more weight to the detection loss and less
to the recognition loss, we will achieve main (detection) - auxiliary
(recognition) multi-task training.

####  4.2.4. Extension: Therapist Response Generation

$cls^{dct}_{emp}$ is effectively asked to forecast the empathy label of a
therapist response absent in the input, which makes the task harder than the
empathy labelling task. To facilitate the learning of the detector, we will
leverage the actual therapist responses in HighRolePlayMI.

For additional input to the baseline, an external open-domain chatbot (e.g.
(Roller et al., 2020)) will be fine-tuned on some counselling (not necessarily
MI) conversation datasets to adapt to the therapeutic domain, and then used
to, for each input, produce response candidates that are appended to the input
in the need-for-empathy detection task.

In a multi-task setting, we can utilise the encoder-decoder backbone of an
open-domain chatbot, but the encoder output is now fed to both 1) an MLP for
need-for-empathy detection, and 2) the decoder for response generation. We
make the detection the main task and the response generation the auxiliary
task in a joint end-to-end training regime, which may allow for insights into
the correlation between the detected need for empathy and the generated
response.

##  5\. Conclusion

We proposed a novel task of detecting client need for empathetic therapist
response. Our plan for tackling this challenge is centred around a labeller-
detector design, where we 1) first train a labeller on empathy-related general
conversations so that it can automatically annotate an unlabelled MI corpus
with binary empathy labels, and 2) then train a detector on the newly labelled
MI corpus to detect real-time client need for empathetic response. We also
laid out our plans for extending the detector with additional-input and multi-
task schemes to improve its ability and explainability. Expecting likely
challenges in our experiments arising from the domain shift from general
conversation to therapy, we hope our proposal and upcoming results will
stimulate more research in empathy-related analysis for clinical counselling.

###### Acknowledgements.

This work has been funded by the EC in the H2020 Marie Skłodowska-Curie
PhilHumans project, contract no. 812882. The authors would also like to thank
Dr. Mark Aloia for his guidance and support.

## References

  * (1)
  * Cao et al. (2019) Jie Cao, Michael Tanana, Zac E. Imel, Eric Poitras, David C. Atkins, and Vivek Srikumar. 2019\.  Observing Dialogue in Therapy: Categorizing and Forecasting Behavioral Codes. In _Proceedings of the 57th Conference of the Association for Computational Linguistics, ACL 2019, Florence, Italy, July 28- August 2, 2019, Volume 1: Long Papers_ , Anna Korhonen, David R. Traum, and Lluís Màrquez (Eds.). Association for Computational Linguistics, 5599–5611.  https://doi.org/10.18653/v1/p19-1563
  * Chen et al. (2019) Zhuohao Chen, Karan Singla, James Gibson, Dogan Can, Zac E. Imel, David C. Atkins, Panayiotis G. Georgiou, and Shrikanth S. Narayanan. 2019.  Improving the Prediction of Therapist Behaviors in Addiction Counseling by Exploiting Class Confusions. In _IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2019, Brighton, United Kingdom, May 12-17, 2019_. IEEE, 6605–6609.  https://doi.org/10.1109/ICASSP.2019.8682885
  * Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019.  BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, NAACL-HLT 2019, Minneapolis, MN, USA, June 2-7, 2019, Volume 1 (Long and Short Papers)_ , Jill Burstein, Christy Doran, and Thamar Solorio (Eds.). Association for Computational Linguistics, 4171–4186.  https://doi.org/10.18653/v1/n19-1423
  * Dridi and Reforgiato Recupero (2019) A. Dridi and D. Reforgiato Recupero. 2019.  Leveraging semantics for sentiment polarity detection in social media.  _International Journal of Machine Learning and Cybernetics_ 10, 8 (2019), 2045–2055.  https://doi.org/10.1007/s13042-017-0727-z cited By 17.
  * Elliott et al. (2011) Robert Elliott, Arthur C Bohart, Jeanne C Watson, and Leslie S Greenberg. 2011.  Empathy.  _Psychotherapy_ 48, 1 (2011), 43.
  * Fleiss (1971) Joseph L Fleiss. 1971\.  Measuring nominal scale agreement among many raters.  _Psychological bulletin_ 76, 5 (1971), 378\.
  * Gibson et al. (2018) James Gibson, David C. Atkins, Torrey Creed, Zac E. Imel, Panayiotis G. Georgiou, and Shrikanth S. Narayanan. 2018.  Multi-label Multi-task Deep Learning for Behavioral Coding.  _CoRR_ abs/1810.12349 (2018).  arXiv:1810.12349 http://arxiv.org/abs/1810.12349
  * Gibson et al. (2016) James Gibson, Dogan Can, Bo Xiao, Zac E. Imel, David C. Atkins, Panayiotis G. Georgiou, and Shrikanth S. Narayanan. 2016\.  A Deep Learning Approach to Modeling Empathy in Addiction Counseling. In _Interspeech 2016, 17th Annual Conference of the International Speech Communication Association, San Francisco, CA, USA, September 8-12, 2016_ , Nelson Morgan (Ed.). ISCA, 1447–1451.  https://doi.org/10.21437/Interspeech.2016-554
  * Gibson et al. (2015) James Gibson, Nikolaos Malandrakis, Francisco Romero, David C. Atkins, and Shrikanth S. Narayanan. 2015\.  Predicting therapist empathy in motivational interviews using language features inspired by psycholinguistic norms. In _INTERSPEECH 2015, 16th Annual Conference of the International Speech Communication Association, Dresden, Germany, September 6-10, 2015_. ISCA, 1947–1951.  http://www.isca-speech.org/archive/interspeech_2015/i15_1947.html
  * Hochreiter and Schmidhuber (1997) Sepp Hochreiter and Jürgen Schmidhuber. 1997.  Long short-term memory.  _Neural computation_ 9, 8 (1997), 1735–1780.
  * Lan et al. (2020) Zhenzhong Lan, Mingda Chen, Sebastian Goodman, Kevin Gimpel, Piyush Sharma, and Radu Soricut. 2020\.  ALBERT: A Lite BERT for Self-supervised Learning of Language Representations. In _8th International Conference on Learning Representations, ICLR 2020, Addis Ababa, Ethiopia, April 26-30, 2020_. OpenReview.net.  https://openreview.net/forum?id=H1eA7AEtvS
  * Li et al. (2020) Shifeng Li, Shi Feng, Daling Wang, Kaisong Song, Yifei Zhang, and Weichao Wang. 2020\.  EmoElicitor: An Open Domain Response Generation Model with User Emotional Reaction Awareness. In _Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI 2020_ , Christian Bessiere (Ed.). ijcai.org, 3637–3643.  https://doi.org/10.24963/ijcai.2020/503
  * Lin et al. (2019) Zhaojiang Lin, Andrea Madotto, Jamin Shin, Peng Xu, and Pascale Fung. 2019.  MoEL: Mixture of Empathetic Listeners. In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing, EMNLP-IJCNLP 2019, Hong Kong, China, November 3-7, 2019_ , Kentaro Inui, Jing Jiang, Vincent Ng, and Xiaojun Wan (Eds.). Association for Computational Linguistics, 121–132.  https://doi.org/10.18653/v1/D19-1012
  * Lin et al. (2020) Zhaojiang Lin, Peng Xu, Genta Indra Winata, Farhad Bin Siddique, Zihan Liu, Jamin Shin, and Pascale Fung. 2020.  CAiRE: An End-to-End Empathetic Chatbot. In _The Thirty-Fourth AAAI Conference on Artificial Intelligence, AAAI 2020, The Thirty-Second Innovative Applications of Artificial Intelligence Conference, IAAI 2020, The Tenth AAAI Symposium on Educational Advances in Artificial Intelligence, EAAI 2020, New York, NY, USA, February 7-12, 2020_. AAAI Press, 13622–13623.  https://aaai.org/ojs/index.php/AAAI/article/view/7098
  * Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019.  RoBERTa: A Robustly Optimized BERT Pretraining Approach.  _CoRR_ abs/1907.11692 (2019).  arXiv:1907.11692 http://arxiv.org/abs/1907.11692
  * Lubis et al. (2018) Nurul Lubis, Sakriani Sakti, Koichiro Yoshino, and Satoshi Nakamura. 2018.  Eliciting Positive Emotion through Affect-Sensitive Dialogue Response Generation: A Neural Network Approach. In _Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, (AAAI-18), the 30th innovative Applications of Artificial Intelligence (IAAI-18), and the 8th AAAI Symposium on Educational Advances in Artificial Intelligence (EAAI-18), New Orleans, Louisiana, USA, February 2-7, 2018_ , Sheila A. McIlraith and Kilian Q. Weinberger (Eds.). AAAI Press, 5293–5300.  https://www.aaai.org/ocs/index.php/AAAI/AAAI18/paper/view/16317
  * Miller and Rollnick (2012) William R Miller and Stephen Rollnick. 2012.  _Motivational interviewing: Helping people change_.  Guilford press.
  * Moyers et al. (2016) Theresa B Moyers, Lauren N Rowell, Jennifer K Manuel, Denise Ernst, and Jon M Houck. 2016.  The motivational interviewing treatment integrity code (MITI 4): rationale, preliminary reliability and validity.  _Journal of substance abuse treatment_ 65 (2016), 36–42.
  * Pérez-Rosas et al. (2017) Verónica Pérez-Rosas, Rada Mihalcea, Kenneth Resnicow, Satinder Singh, and Lawrence An. 2017\.  Understanding and Predicting Empathic Behavior in Counseling Therapy. In _Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics, ACL 2017, Vancouver, Canada, July 30 - August 4, Volume 1: Long Papers_ , Regina Barzilay and Min-Yen Kan (Eds.). Association for Computational Linguistics, 1426–1435.  https://doi.org/10.18653/v1/P17-1131
  * Pérez-Rosas et al. (2019) Verónica Pérez-Rosas, Xinyi Wu, Kenneth Resnicow, and Rada Mihalcea. 2019.  What Makes a Good Counselor? Learning to Distinguish between High-quality and Low-quality Counseling Conversations. In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_. Association for Computational Linguistics, Florence, Italy, 926–935.  https://doi.org/10.18653/v1/P19-1088
  * Radford et al. (2018) Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever. 2018.  Improving language understanding by generative pre-training (2018).  _URL https://s3-us-west-2. amazonaws. com/openai-assets/research-covers/language-unsupervised/language_ understanding_paper. pdf_ (2018).
  * Radford et al. (2019) Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. 2019\.  Language models are unsupervised multitask learners.  _OpenAI Blog_ 1, 8 (2019), 9.
  * Rashkin et al. (2019) Hannah Rashkin, Eric Michael Smith, Margaret Li, and Y-Lan Boureau. 2019.  Towards Empathetic Open-domain Conversation Models: A New Benchmark and Dataset. In _Proceedings of the 57th Conference of the Association for Computational Linguistics, ACL 2019, Florence, Italy, July 28- August 2, 2019, Volume 1: Long Papers_ , Anna Korhonen, David R. Traum, and Lluís Màrquez (Eds.). Association for Computational Linguistics, 5370–5381.  https://doi.org/10.18653/v1/p19-1534
  * Recupero et al. (2014) D.R. Recupero, S. Consoli, A. Gangemi, A.G. Nuzzolese, and D. Spampinato. 2014.  A semantic web based core engine to efficiently perform sentiment analysis.  _Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)_ 8798 (2014), 245–248.  https://doi.org/10.1007/978-3-319-11955-7_28 cited By 17.
  * Recupero et al. (2015) D.R. Recupero, M. Dragoni, and V. Presutti. 2015\.  ESWC 15 challenge on concept-level sentiment analysis.  _Communications in Computer and Information Science_ 548 (2015), 211–222.  https://doi.org/10.1007/978-3-319-25518-7_18 cited By 17.
  * Recupero et al. (2019) Diego Reforgiato Recupero, Mehwish Alam, Davide Buscaldi, Aude Grezka, and Farideh Tavazoee. 2019\.  Frame-Based Detection of Figurative Language in Tweets [Application Notes].  _IEEE Comput. Intell. Mag._ 14, 4 (2019), 77–88.  https://doi.org/10.1109/MCI.2019.2937614
  * Reforgiato Recupero and Cambria (2014) D. Reforgiato Recupero and E. Cambria. 2014.  ESWC’14 challenge on Concept-Level Sentiment Analysis.  _Communications in Computer and Information Science_ 475 (2014), 3–20.  https://doi.org/10.1007/978-3-319-12024-9_1 cited By 17.
  * Roller et al. (2020) Stephen Roller, Emily Dinan, Naman Goyal, Da Ju, Mary Williamson, Yinhan Liu, Jing Xu, Myle Ott, Kurt Shuster, Eric Michael Smith, Y-Lan Boureau, and Jason Weston. 2020\.  Recipes for building an open-domain chatbot.  _CoRR_ abs/2004.13637 (2020).  arXiv:2004.13637 https://arxiv.org/abs/2004.13637
  * Shin et al. (2020) Jamin Shin, Peng Xu, Andrea Madotto, and Pascale Fung. 2020\.  Generating Empathetic Responses by Looking Ahead the User’s Sentiment. In _2020 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2020, Barcelona, Spain, May 4-8, 2020_. IEEE, 7989–7993.  https://doi.org/10.1109/ICASSP40776.2020.9054379
  * Singla et al. (2020) Karan Singla, Zhuohao Chen, David C. Atkins, and Shrikanth Narayanan. 2020.  Towards end-2-end learning for predicting behavior codes from spoken utterances in psychotherapy conversations. In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, ACL 2020, Online, July 5-10, 2020_ , Dan Jurafsky, Joyce Chai, Natalie Schluter, and Joel R. Tetreault (Eds.). Association for Computational Linguistics, 3797–3803.  https://www.aclweb.org/anthology/2020.acl-main.351/
  * Sordoni et al. (2015) Alessandro Sordoni, Yoshua Bengio, Hossein Vahabi, Christina Lioma, Jakob Grue Simonsen, and Jian-Yun Nie. 2015.  A Hierarchical Recurrent Encoder-Decoder for Generative Context-Aware Query Suggestion. In _Proceedings of the 24th ACM International Conference on Information and Knowledge Management, CIKM 2015, Melbourne, VIC, Australia, October 19 - 23, 2015_ , James Bailey, Alistair Moffat, Charu C. Aggarwal, Maarten de Rijke, Ravi Kumar, Vanessa Murdock, Timos K. Sellis, and Jeffrey Xu Yu (Eds.). ACM, 553–562.  https://doi.org/10.1145/2806416.2806493
  * Tanana et al. (2016) Michael Tanana, Kevin A Hallgren, Zac E Imel, David C Atkins, and Vivek Srikumar. 2016.  A comparison of natural language processing methods for automated coding of motivational interviewing.  _Journal of substance abuse treatment_ 65 (2016), 43–50.
  * Wang et al. (2019) Alex Wang, Amanpreet Singh, Julian Michael, Felix Hill, Omer Levy, and Samuel R. Bowman. 2019\.  GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding. In _7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019_. OpenReview.net.  https://openreview.net/forum?id=rJ4km2R5t7
  * Xiao et al. (2014) Bo Xiao, Daniel Bone, Maarten Van Segbroeck, Zac E. Imel, David C. Atkins, Panayiotis G. Georgiou, and Shrikanth S. Narayanan. 2014\.  Modeling therapist empathy through prosody in drug addiction counseling. In _INTERSPEECH 2014, 15th Annual Conference of the International Speech Communication Association, Singapore, September 14-18, 2014_ , Haizhou Li, Helen M. Meng, Bin Ma, Engsiong Chng, and Lei Xie (Eds.). ISCA, 213–217.  http://www.isca-speech.org/archive/interspeech_2014/i14_0213.html
  * Xiao et al. (2012) Bo Xiao, Dogan Can, Panayiotis G. Georgiou, David C. Atkins, and Shrikanth S. Narayanan. 2012\.  Analyzing the language of therapist empathy in Motivational Interview based psychotherapy. In _Asia-Pacific Signal and Information Processing Association Annual Summit and Conference, APSIPA 2012, Hollywood, CA, USA, December 3-6, 2012_. IEEE, 1–4.  http://ieeexplore.ieee.org/document/6411762/
  * Xiao et al. (2016a) Bo Xiao, Dogan Can, James Gibson, Zac E. Imel, David C. Atkins, Panayiotis G. Georgiou, and Shrikanth S. Narayanan. 2016a.  Behavioral Coding of Therapist Language in Addiction Counseling Using Recurrent Neural Networks. In _Interspeech 2016, 17th Annual Conference of the International Speech Communication Association, San Francisco, CA, USA, September 8-12, 2016_ , Nelson Morgan (Ed.). ISCA, 908–912.  https://doi.org/10.21437/Interspeech.2016-1560
  * Xiao et al. (2016b) Bo Xiao, Che-Wei Huang, Zac E. Imel, David C. Atkins, Panayiotis G. Georgiou, and Shrikanth S. Narayanan. 2016b.  A technology prototype system for rating therapist empathy from audio recordings in addiction counseling.  _PeerJ Comput. Sci._ 2 (2016), e59.  https://doi.org/10.7717/peerj-cs.59
  * Xiao et al. (2015) Bo Xiao, Zac E. Imel, David C. Atkins, Panayiotis G. Georgiou, and Shrikanth S. Narayanan. 2015\.  Analyzing speech rate entrainment and its relation to therapist empathy in drug addiction counseling. In _INTERSPEECH 2015, 16th Annual Conference of the International Speech Communication Association, Dresden, Germany, September 6-10, 2015_. ISCA, 2489–2493.  http://www.isca-speech.org/archive/interspeech_2015/i15_2489.html
  * Zandie and Mahoor (2020) Rohola Zandie and Mohammad H. Mahoor. 2020.  EmpTransfo: A Multi-Head Transformer Architecture for Creating Empathetic Dialog Systems. In _Proceedings of the Thirty-Third International Florida Artificial Intelligence Research Society Conference, Originally to be held in North Miami Beach, Florida, USA, May 17-20, 2020_ , Roman Barták and Eric Bell (Eds.). AAAI Press, 276–281.  https://aaai.org/ocs/index.php/FLAIRS/FLAIRS20/paper/view/18446
  * Zhong et al. (2020) Peixiang Zhong, Yan Sun, Yong Liu, Chen Zhang, Hao Wang, Zaiqing Nie, and Chunyan Miao. 2020.  Endowing Empathetic Dialogue Systems with Personas.  _CoRR_ abs/2004.12316 (2020).  arXiv:2004.12316 https://arxiv.org/abs/2004.12316
  * Zhou and Wang (2018) Xianda Zhou and William Yang Wang. 2018.  MojiTalk: Generating Emotional Responses at Scale. In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics, ACL 2018, Melbourne, Australia, July 15-20, 2018, Volume 1: Long Papers_ , Iryna Gurevych and Yusuke Miyao (Eds.). Association for Computational Linguistics, 1128–1137.  https://doi.org/10.18653/v1/P18-1104


# Floquet-Anderson localization in the Thouless pump and how to avoid it

András Grabarits  Department of Theoretical Physics, Institute of Physics,
Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111
Budapest, Hungary  MTA-BME Quantum Dynamics and Correlations Research Group,
Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111
Budapest, Hungary  Department of Physics and Materials Science, University of
Luxembourg, L-1511 Luxembourg, Luxembourg  Attila Takács  Department of
Theoretical Physics, Institute of Physics, Budapest University of Technology
and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary  Université de
Lorraine, CNRS, LPCT, F-54000 Nancy, France  Ion Cosma Fulga  Leibniz
Institute for Solid State and Materials Research, IFW Dresden,
Helmholtzstrasse 20, 01069 Dresden, Germany  Würzburg-Dresden Cluster of
Excellence ct.qmat, 01062 Dresden, Germany  János K. Asbóth  Department of
Theoretical Physics, Institute of Physics, Budapest University of Technology
and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary  Wigner Research
Centre for Physics, H-1525 Budapest, P.O. Box 49., Hungary

###### Abstract

We investigate numerically how onsite disorder affects conduction in the
periodically driven Rice-Mele model, a prototypical realization of the
Thouless pump. Although the pump is robust against disorder in the fully
adiabatic limit, much less is known about the case of finite period time $T$,
which is relevant also in light of recent experimental realizations. We find
that at any fixed period time and nonzero disorder, increasing the system size
$L\to\infty$ always leads to a breakdown of the pump, indicating Anderson
localization of the Floquet states. Our numerics indicate, however, that in a
properly defined thermodynamic limit, where $L/T^{\theta}$ is kept constant,
Anderson localization can be avoided, and the charge pumped per cycle has a
well-defined value – as long as the disorder is not too strong. The critical
exponent $\theta$ is not universal, rather, its value depends on the disorder
strength. Our findings are relevant for practical, experimental realizations
of the Thouless pump, for studies investigating the nature of its current-
carrying Floquet eigenstates, as well as the mechanism of the full breakdown
of the pump, expected if the disorder exceeds a critical value.

The Thouless pump [1] was instrumental in understanding the role topology
plays in the theory of the quantum Hall effect. Its simplest form is that of a
two-band, gapped fermionic chain whose parameters are slowly and periodically
varied in time. In the half-filled state, where the lower/upper band is
completely filled/empy, and in the adiabatic limit, an integer number $Q$ of
fermions are pumped in the lower band, where $Q$ is a topological invariant,
equal to the Chern number $C$ of the pump sequence.

While it started out as a thought experiment, the Thouless pump can now be
found in the lab [2]. After its demonstration in photonic systems [3, 4, 5,
6], topological pumping has been realized in a variety of platforms, such as
mechanical metamaterials [7, 8], ultracold atoms [9, 10, 11], and other
quantum systems [12]. More recently, an electrical circuit demonstration has
been put forward [13].

The real-life Thouless pump has a finite size, $L$, a finite period time, $T$,
and it is disordered. This raises the question whether these, either
separately or in combination, can prove detrimental to its robustness, namely
to the quantization of the pumped charge. For instance, even without disorder,
a finite period time gives corrections to the quantized value of the pumped
charge. For a quench-like switching on of the pump, Ref. [14] found these to
scale as $1/T^{2}$, but they should be greatly reduced for a smoother
switching-on of the periodic driving cycle. The breaking of adiabaticity,
however, is not always sufficient to destroy the Thouless pump. As shown in
Ref. [15], if the disorder-free pump is made longer and longer, then
adiabaticity will be strongly broken for any finite $T$, no matter how large.
In spite of this, they found that the quantization of the pumped charge
survives in the steady-state regime, when the pump performs cycle after cycle.
Similarly, in the adiabatic $T\to\infty$ limit, the quantization of the pumped
charge should be robust against small disorder, as discussed by Thouless and
Niu [16]. Here, eigenstates of the time-dependent Hamiltonian are all Anderson
localized, and in the adiabatic limit an intuitive (although possibly
misleading) picture is that periodic modulation pumps charge between them.

In this Letter we focus on the effect of onsite disorder on an actual Thouless
pump (finite $T$ rather than the adiabatic limit). On the one hand, disorder
can even result in a suppression of finite-$T$ corrections, and a higher
pumped charge [17]. On the other, adding too much disorder has to result in a
breakdown of the pump, via an Anderson localization transition – a few works
have already studied this numerically [17, 18].

Onsite disorder on the Thouless pump is particularly interesting because of
the connection to the “levitation and annihilation” in Chern insulators [19].
Disorder in the Chern insulator localizes its eigenstates, but each
topological band has (at least) one state that remains extended in the
thermodynamic limit, which “carries the Chern number” [20, 21, 22]. As
disorder is increased, full Anderson localization happens by these robustly
extended states “levitating” towards each other in energy and “annihilating”.
Can such phenomena be observed in the Floquet states of the disordered
Thouless pump? Numerical results [17] are consistent with this, and have even
identified a critical exponent for this Anderson localization transition,
obtained by scaling up the size of the pump at a constant (and large) period
time.

There is an issue with the disordered Thouless pump in the thermodynamic
limit, however, that to the best of our knowledge has not been directly
addressed yet. One might think that for a thermodynamic limit, $L\to\infty$
and $T\to\infty$ should be taken one after the other. However, we argue this
is inadequate. Taking $T\to\infty$ first is problematic, since the charge pump
becomes infinitely slow. Moreover, in this “ultra-adiabatic” limit transitions
occur between distant Anderson localized eigenstates, thus computation of $Q$
needs open boundary conditions [23]. Taking $L\to\infty$ first is often
(sometimes tacitly) assumed. However, as we show in the following, this leads
to a breakdown of the pump due to the Anderson localization of the Floquet
eigenstates. In this Letter we suggest a properly defined way to take
$L\to\infty$ and $T\to\infty$.

_The model._ — We consider the periodically driven Rice-Mele model [24] with
an onsite potential disorder that is independent of time. Spinless fermions
hop on a closed chain of $L=2N$ sites, with the unitary time evolution
governed by the Hamiltonian,

$\displaystyle\hat{H}(t)$
$\displaystyle=-\sum_{m=1}^{L}\left[J+(-1)^{m}\tilde{J}\cos\tfrac{2\pi
t}{T}\right]\hat{c}^{\dagger}_{m}\hat{c}_{m+1}+\mathrm{h.c}.$
$\displaystyle-\sum_{m=1}^{L}\left[(-1)^{m}\Delta\sin\tfrac{2\pi
t}{T}+W\zeta_{m}\right]\hat{c}^{\dagger}_{m}\hat{c}_{m},$ (1)

where $\hat{c}_{m}$ annihilates a fermion on site $m$, with
$\hat{c}_{L+1}=\hat{c}_{1}$, i.e., periodic boundary conditions,
$J$/$\tilde{J}$ are uniform/staggered components of the nearest-neighbor
hopping, $\Delta$ is a staggered onsite potential, and $t$ and $T$ are time
and period time. The onsite disorder has amplitude $W$ and the $\zeta_{m}$’s
are real random numbers uniformly distributed on $[-1/2,1/2]$. We set
$\hbar=1$ for convenience. In this noninteracting model, all quantitites of
interest can be computed from the single-particle $L\times L$ Hamiltonian
matrix $H(t)$, with
$\hat{H}(t)=\sum_{l,m=1}^{L}\hat{c}_{l}^{\dagger}H_{lm}(t)\hat{c}_{m}$.

We use the basis of Floquet states: eigenstates $\ket{\psi_{n}}$ of the
single-particle (Floquet) unitary operator $\hat{U}$ for one period of time
evolution, $\hat{U}\ket{\psi_{n}}=e^{-i\varepsilon_{n}}\ket{\psi_{n}}$. Here
$\hat{U}=\mathcal{T}e^{-i\int_{0}^{T}\mathrm{d}t\hat{H}(t)}$, where ${\cal T}$
is time ordering, $n=1,2,\ldots,L$ is the eigenstate index and
$\varepsilon_{n}$ is the quasienergy. Floquet states evolve periodically in
time, up to a phase factor:

$\displaystyle\ket{\psi_{n}(t)}$
$\displaystyle=\mathcal{T}\,e^{-i\int_{0}^{t}\mathrm{d}t^{\prime}\hat{H}(t^{\prime})}\ket{\psi_{n}}=e^{i\varepsilon_{n}T}\ket{\psi_{n}(t+T)}.$
(2)

If the disorder is weak and the pump is run slowly enough, Floquet states can
be assigned to bands according to their average energy,

$\displaystyle\overline{E_{n}}$
$\displaystyle=\frac{1}{T}\int_{0}^{T}\mathrm{d}t\bra{\psi_{n}(t)}\hat{H}(t)\ket{\psi_{n}(t)}.$
(3)

Floquet states carry current, whose integral over the time period gives the
pumped charge in that state,

$\displaystyle Q_{n}$
$\displaystyle=2\int_{0}^{T}\mathrm{d}t\left(J+\tilde{J}\cos\tfrac{2\pi
t}{T}\right)\mathrm{Im}[\psi_{n,2}^{\ast}(t)\psi_{n,1}(t)].$ (4)

Here we take the current between sites 1 and 2, but the position does not
matter, due to the periodicity of the time evolution of Floquet states.

We calculate the charge pumped in the so-called sustained pumping limit of a
filled lower band [25, 17]: The system is initialized at $t=0$, with the $L/2$
lowest energy eigenstates $\ket{\phi_{l}}$ of the instantaneous Hamiltonian
fully occupied, and then is time evolved. After many cycles, this results
effectively in a Floquet diagonal ensemble [25], i.e., an incoherent mixture
where Floquet states are populated with the same weights as at $t=0$. Thus,
the charge pumped per cycle in this limit is

$\displaystyle Q$
$\displaystyle=\sum_{n=1}^{L}Q_{n}\sum_{l=1}^{L/2}\left|{\bra{\phi_{l}}\psi_{n}\rangle}\right|^{2}.$
(5)

_The numerical method._ — We compute the time evolution of the Floquet states,
needed for Eq. (4), as a matrix product of time-slices of the timestep
operator. For the time-slices, we used a recently-developed method based on
the Chebyshev polynomial representation of skew Hermitian matrices,
$e^{-iH\mathrm{d}t}\approx\alpha_{0}-iz_{0}H\mathrm{d}t-\alpha_{1}[H\mathrm{d}t]^{2}+iz_{1}[H\mathrm{d}t]^{3}+\alpha_{2}[H\mathrm{d}t]^{4}-iz_{2}[H\mathrm{d}t]^{5}$,
with constants specified to 20 decimals [26]. This gives the matrix
exponential to numerical accuracy, as long as $\lvert\lvert
H(t)\mathrm{d}t\rvert\rvert_{1}<1.17\times 10^{-2}$. We could reach chains
lengths up to $L=10000$, more than a factor of 10 larger than previous works
[17]. We use the hopping $J$ as our energy scale, and set parameters as

$\displaystyle J$ $\displaystyle=1;$ $\displaystyle\tilde{J}$
$\displaystyle=1/2;$ $\displaystyle\Delta$ $\displaystyle=1.5,$ (6)

for a well-defined gap with a Chern number $C=1$. Thus, $Q=1$ in the adiabatic
limit, as long as the instantaneous Hamiltonian is gapped, i.e., $W\lesssim
3.5$ [17].

Figure 1:  Pumped charge $Q$ decreases as chain length $L$ is increased from
$80$ to $4480$, with disorder $W=2.5$ (average of 20 disorder realizations).
Results for various period times $T$ fall onto each other if rescaling the
system size as $L/{T^{\theta}}$, with $\theta=3.95$. Inset: unscaled data.

_Results._ — We found that the disordered Thouless pump breaks down when
increasing the length $L$ of the chain, keeping the period time $T$ constant.
Examples are shown in Fig. 1, for disorder $W=2.5$, for $T=8$ to $50$, and
$L=80$ up to $4480$ In these and all cases we studied, the pumped charge
decreased as the length was increased. This suggests that Floquet-Anderson
localization does not only set in when the disorder is large ($W>3.5$), but
occurs for any onsite disorder, $W>0$. The length $L$ where the pumped charge
decreases significantly (e.g., $Q=1/2$, or $Q=1/4$) provides an estimate for
the Floquet localization length $\zeta_{F}$.

We found that slower driving (longer period times $T$) leads to more resilient
Thouless pumps, with an apparent power-law relation

$\displaystyle Q(L,T,W)$
$\displaystyle=Q(L^{\prime},(L^{\prime}/L)^{1/\theta(W)}T,W).$ (7)

This is suggested by the good collapse of the numerically measured $Q$ values
when using the above scaling relation, as shown in Fig. 1. Thus, the Floquet
localization length appears to scale with the period time as $\zeta_{F}\propto
T^{\theta}$.

Figure 2:  The scaling exponent $\theta$, obtained by three numerical
approaches for each value of disorder $W$ (see main text for details). The
exponent decreases as $W$ increases, while it appears to diverge at $W=0$,
consistent with the pumped charge $Q$ being independent of the system size in
the clean case – this latter is shown in the inset. For a detailed description
of how the exponent was extracted see the Supplemental Material [27].

We found that the exponent $\theta$ of Eq. (7) does not take on a universal
value, but depends continuously on the disorder $W$, as shown in Fig. 2. We
extracted the exponent by three different methods. First and second, by
identifying $\zeta_{F}$ with the system size where the pumped charge is
$Q=1/2$, and $Q=1/4$, respectively. Third, we took all the data for a fixed
$W$, and various $T$ and $L$ values, and fitted it with a three-parameter
Ansatz - detailed in the Supplemental Material (SM, [27]). These methods
agree, and give a disorder-dependent exponent $\theta$, which approaches
$\theta\approx 2$ near the critical disorder $W\approx 3.5$. Note, however,
that the numerical evidence for the power-law scaling is strong only for the
case of moderate disorder. For smaller disorders, $W\lesssim 1.5$, we have
$1/\theta\ll 1$, thus in the numerically available range the evidence for the
power-law scaling here is not conclusive, as discussed in the SM [27].

Figure 3:  Colormap: Pumped charge $Q$, for $L=320$, for various period times
$T$ and disorders $W$ (average over 20 disorder realizations). The Thouless
pump works well (light area) for small $W$ and large $T$, and breaks down both
if $T$ decreases or $W$ increases. For $W\gg W_{c}\approx 3.5$, where there is
no gap in the instantaneous energy spectrum, the pumped charge is very close
to 0 for any $T$. Black lines: $T(W)$ curves along which $Q=1/4$. These show
that the pump breaks down easier if the chain is longer.

For a more complete picture of the breakdown of the Thouless pump as a
function of disorder $W$, chain length $L$, and period time $T$, we show the
numerically obtained map of pumped charge $Q$ in Fig. 3. The colors show $Q$
values for $L=320$, and results for other lengths are shown as $Q=1/4$
isolines. These reveal four qualitatively different regimes of the charge
pump. For small disorder, $W\lesssim 0.5$, the Floquet localization length
$\zeta_{F}$ decreases sharply as the disorder is increased. For $0.5\lesssim
W\lesssim 2$, and period times $10\lesssim T\lesssim 20$, $\zeta_{F}$ does not
depend much on the disorder strength. For larger disorder, $2\lesssim
W\lesssim W_{c}=3.5$, we have a sharp decrease of the $\zeta_{F}$ as $W$ is
increased. Finally, above the critical disorder value, $3.5\lesssim W$, we
observe the charge pump breaking down completely.

Figure 4:  Spectrum of the average energy of Floquet states in a pump with
disorder $W=2$, for various system lengths, and with color representing the
IPR value of the states (single disorder realizations). In (a), the period
time fixed, $T=20$, increasing the length leads to a qualitative change in the
spectrum: For $L\lesssim 100$, two bands of extended states can be seen, which
carry current right/left in the lower/upper band; for larger lengths the gap
between the bands is closed. In (b), the period time is increased together
with the system size, keeping $L/T^{\theta}$ constant, where $\theta={6.01}$
was used. Here there is no qualitative change in the spectrum as $L$ is
increased, however, the range of extended states becomes narrower.

_Thermodynamic limit._ — For a deeper understanding of how disorder impacts
the Thouless pump, we define an alternate thermodynamic limit: $L\to\infty$
and $T\to\infty$ together, with $L/T^{\theta}$ kept constant. This is needed,
e.g., to explore the extended/localized nature of the Floquet states, using
the inverse participation ratio [17] (IPR)
$P_{2}=\sum_{m=1}^{L}\left|\psi_{n,m}\right|^{4}$. To show how this limit
avoids the problem of the Anderson localization of Floquet states, see Fig. 4.
We choose parameters so that for short lengths, $L\lesssim 100$, the Thouless
pump works well, and Floquet states form two well separated bands. If the
length is increased at fixed $T$ (panel a), the two bands merge and the pump
begins to break down. In contrast, if $L/T^{\theta}$ is kept constant (panel
b), the spectrum of Floquet states shows no qualitative change up to the
largest system sizes that we were able to access numerically. For each band,
states in the band center are more extended (lower IPR, decreasing with $L$)
and at the band edges more localized (higher IPR, independent of $L$). More
detailed analysis of the IPR values, and examples for shorter period times
$T$, where the breakdown of the pump is even more visible, are shown in the SM
[27].

_Discussion, conclusions._ — We found that onsite potential disorder in the
periodically driven Rice-Mele model results in a breakdown of the Thouless
pump in the $L\to\infty$, constant-$T$ limit, due to the Anderson localization
of the Floquet states. This can be avoided by taking $L\to\infty$ and
$T\to\infty$ together, keeping $L/T^{\theta}$ constant, where $\theta$ is a
disorder-dependent critical exponent. Although we expected $\theta=2$ based on
the corrections to adiabaticity, we found that this is not the case, rather,
$\theta$ depends on disorder strength continuously. It is an interesting open
problem to find an analytical explanation of this phenomenon.

Our work is a starting point for a more systematic investigation of the
relation between Anderson localization in the Thouless pump and the
“levitation and annihilation” of extended states in Chern insulators. By
finding a suitable thermodynamic limit, we open the way to studying
numerically the conduction in the disordered Thouless pump, as well as the
mechanism of its breakdown as disorder is increased. Our preliminary results,
shown in the SM [27], suggest that the current-carrying states here have a
fractal nature, but it is an open question whether in the thermodynamic it is
only a single Floquet state per band that carries the current.

From a broader perspective, the Thouless pump is the oldest among a large
family of topological pumps, which by now go well beyond Chern insulators.
Topological pumping has been associated with a wide range of topological
insulators and superconductors [28, 29, 30], and has been proposed to occur
also between topological defects [31, 32]. It has been further extended to
higher-order topological phases [33, 34, 35], which can lead to dipole or to
quadrupole pumps [36]. Very recently, topological pumping between the corners
of a two-dimensional sample has been shown to occur both theoretically and
experimentally, with the pump working either via bulk states [37, 38], or via
edge states [39]. Our work opens a new direction of research in this field,
consisting in the study of the thermodynamic limits associated to this large
family of pumps and the critical exponents characterizing them.

_Acknowledgements_ We acknowledge support from the National Research,
Development and Innovation Office (NKFIH) through the OTKA research grants No.
K138606 and 132146, and within the Quantum National Laboratory of Hungary
program (Grant No. 2022-2.1.1-NL-2022-00004). ICF acknowledges support from
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy through the Würzburg-Dresden Cluster of
Excellence on Complexity and Topology in Quantum Matter – _ct.qmat_ (EXC 2147,
project-ids 390858490 and 392019).

## References

  * Thouless [1983] D. J. Thouless, Quantization of particle transport, Phys. Rev. B 27, 6083 (1983).
  * Citro and Aidelsburger [2023] R. Citro and M. Aidelsburger, Thouless pumping and topology, Nat. Rev. Phys. 5, 87 (2023).
  * Kraus  _et al._ [2012] Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Topological states and adiabatic pumping in quasicrystals, Phys. Rev. Lett. 109, 106402 (2012).
  * Verbin  _et al._ [2015] M. Verbin, O. Zilberberg, Y. Lahini, Y. E. Kraus, and Y. Silberberg, Topological pumping over a photonic Fibonacci quasicrystal, Phys. Rev. B 91, 064201 (2015).
  * Ke  _et al._ [2016] Y. Ke, X. Qin, F. Mei, H. Zhong, Y. S. Kivshar, and C. Lee, Topological phase transitions and Thouless pumping of light in photonic waveguide arrays, Laser Photonics Rev. 10, 995 (2016).
  * Cerjan  _et al._ [2020] A. Cerjan, M. Wang, S. Huang, K. P. Chen, and M. C. Rechtsman, Thouless pumping in disordered photonic systems, Light Sci. Appl. 9, 178 (2020).
  * Grinberg  _et al._ [2020] I. H. Grinberg, M. Lin, C. Harris, W. A. Benalcazar, C. W. Peterson, T. L. Hughes, and G. Bahl, Robust temporal pumping in a magneto-mechanical topological insulator, Nat. Commun. 11, 974 (2020).
  * Xia  _et al._ [2021] Y. Xia, E. Riva, M. I. Rosa, G. Cazzulani, A. Erturk, F. Braghin, and M. Ruzzene, Experimental observation of temporal pumping in electromechanical waveguides, Phys. Rev. Lett. 126, 095501 (2021).
  * Lohse  _et al._ [2015] M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice, Nature Phys. 12, 350 (2015).
  * Nakajima  _et al._ [2016] S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, Topological Thouless pumping of ultracold fermions, Nature Phys. 12, 296 (2016).
  * Lu  _et al._ [2016] H.-I. Lu, M. Schemmer, L. Aycock, D. Genkina, S. Sugawa, and I. Spielman, Geometrical pumping with a Bose-Einstein condensate, Phys. Rev. Lett. 116, 200402 (2016).
  * Ma  _et al._ [2018] W. Ma, L. Zhou, Q. Zhang, M. Li, C. Cheng, J. Geng, X. Rong, F. Shi, J. Gong, and J. Du, Experimental observation of a generalized thouless pump with a single spin, Phys. Rev. Lett. 120, 120501 (2018).
  * Stegmaier  _et al._ [2023] A. Stegmaier, H. Brand, S. Imhof, A. Fritzsche, T. Helbig, T. Hofmann, I. Boettcher, M. Greiter, C. H. Lee, G. Bahl, A. Szameit, T. Kießling, R. Thomale, and L. K. Upreti, Realizing efficient topological temporal pumping in electrical circuits, arXiv:2306.15434 (2023).
  * Privitera  _et al._ [2018] L. Privitera, A. Russomanno, R. Citro, and G. E. Santoro, Nonadiabatic breaking of topological pumping, Phys. Rev. Lett. 120, 106601 (2018).
  * Lychkovskiy  _et al._ [2017] O. Lychkovskiy, O. Gamayun, and V. Cheianov, Time scale for adiabaticity breakdown in driven many-body systems and orthogonality catastrophe, Phys. Rev. Lett. 119, 200401 (2017).
  * Niu and Thouless [1984] Q. Niu and D. J. Thouless, Quantised adiabatic charge transport in the presence of substrate disorder and many-body interaction, J. Phys. A: Math. Gen. 17, 2453 (1984).
  * Wauters  _et al._ [2019] M. M. Wauters, A. Russomanno, R. Citro, G. E. Santoro, and L. Privitera, Localization, topology, and quantized transport in disordered Floquet systems, Phys. Rev. Lett. 123, 266601 (2019).
  * Hayward  _et al._ [2021] A. L. C. Hayward, E. Bertok, U. Schneider, and F. Heidrich-Meisner, Effect of disorder on topological charge pumping in the Rice-Mele model, Phys. Rev. A 103, 043310 (2021).
  * Laughlin [1984] R. B. Laughlin, Levitation of extended-state bands in a strong magnetic field, Phys. Rev. Lett. 52, 2304 (1984).
  * Halperin [1982] B. I. Halperin, Quantized Hall conductance, current-carrying edge states, and the existence of extended states in a two-dimensional disordered potential, Phys. Rev. B 25, 2185 (1982).
  * Thouless [1984] D. J. Thouless, Wannier functions for magnetic sub-bands, J. Phys. C: Solid State Phys. 17, L325 (1984).
  * Thonhauser and Vanderbilt [2006] T. Thonhauser and D. Vanderbilt, Insulator/Chern-insulator transition in the Haldane model, Phys. Rev. B 74, 235111 (2006).
  * Imura  _et al._ [2018] K.-I. Imura, Y. Yoshimura, T. Fukui, and Y. Hatsugai, Bulk-edge correspondence in topological transport and pumping, J. Phys.: Conf. Ser. 969, 012133 (2018).
  * Rice and Mele [1982] M. J. Rice and E. J. Mele, Elementary excitations of a linearly conjugated diatomic polymer, Phys. Rev. Lett. 49, 1455 (1982).
  * Russomanno  _et al._ [2012] A. Russomanno, A. Silva, and G. E. Santoro, Periodic steady regime and interference in a periodically driven quantum system, Phys. Rev. Lett. 109, 257201 (2012).
  * Bader  _et al._ [2022] P. Bader, S. Blanes, F. Casas, and M. Seydaoğlu, An efficient algorithm to compute the exponential of skew-Hermitian matrices for the time integration of the Schrödinger equation, Math. Comput. Simul. 194, 383 (2022).
  * [27] In the Supplemental Material, we show the details about the methods to determine the exponents, describe how we could reach large system sizes in the clean case and provide additional information about the localization properties of the Floquet states .
  * Meidan  _et al._ [2011] D. Meidan, T. Micklitz, and P. W. Brouwer, Topological classification of adiabatic processes, Phys. Rev. B 84, 195410 (2011).
  * Ryu  _et al._ [2012] S. Ryu, J. E. Moore, and A. W. W. Ludwig, Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors, Phys. Rev. B 85, 045104 (2012).
  * Fulga  _et al._ [2012] I. C. Fulga, F. Hassler, and A. R. Akhmerov, Scattering theory of topological insulators and superconductors, Phys. Rev. B 85, 165409 (2012).
  * Teo and Kane [2010] J. C. Y. Teo and C. L. Kane, Topological defects and gapless modes in insulators and superconductors, Physical Review B 82, 115120 (2010).
  * Xie  _et al._ [2022] B.-Y. Xie, O. You, and S. Zhang, Photonic topological pump between chiral disclination states, Phys. Rev. A 106, l021502 (2022).
  * Benalcazar  _et al._ [2017a] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Quantized electric multipole insulators, Science 357, 61 (2017a).
  * Song  _et al._ [2017] Z. Song, Z. Fang, and C. Fang, $(d-2)$-dimensional edge states of rotation symmetry protected topological states, Phys. Rev. Lett. 119, 246402 (2017).
  * Schindler  _et al._ [2018] F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, Higher-order topological insulators, Sci. Adv. 4, aat0346 (2018).
  * Benalcazar  _et al._ [2017b] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators, Phys. Rev. B 96, 245115 (2017b).
  * Wienand  _et al._ [2022] J. F. Wienand, F. Horn, M. Aidelsburger, J. Bibo, and F. Grusdt, Thouless pumps and bulk-boundary correspondence in higher-order symmetry-protected topological phases, Phys. Rev. Lett. 128, 246602 (2022).
  * Benalcazar  _et al._ [2022] W. A. Benalcazar, J. Noh, M. Wang, S. Huang, K. P. Chen, and M. C. Rechtsman, Higher-order topological pumping and its observation in photonic lattices, Phys. Rev. B 105, 195129 (2022).
  * Xia  _et al._ [2023] T. Xia, Y. Li, Q. Zhang, X. Fan, M. Xiao, and C. Qiu, Observation of hybrid-order topological pump in a Kekule-textured graphene lattice, arXiv:2309.04051 (2023).