# A brief perspective of high temperature superconductivity in the cuprates:
Strong correlations combined with superexchange match experiment

Józef Spałek  Institute of Theoretical Physics, Jagiellonian University, ul.
Łojasiewicza 11, PL-30-348 Kraków, Poland<EMAIL_ADDRESS>
###### Abstract

High temperature superconductivity encompasses the cuprates, nickelates, iron
pnictides, and LaHx compounds. The first three groups of compounds involve in
the pairing electrons, which are strongly to moderately correlated, whereas in
the last class of systems specific phonon excitations. In this overview we
concentrate first on the (semi)quantitative theory of high TC
superconductivity in the cuprates based on our original vibrational approach
beyond the renormalized mean field theory. The model we explore mainly is
$t$-$J$-$U$ model containing both the superexchange (kinetic energy) combined
with strong interelectronic correlations. Selected equilibrium and dynamic-
excitation properties are analyzed briefly. General questions regarding the
pseudogap and two–dimensional character of those systems are raised.

††preprint: APS/123-QED

##  I Introduction

Originally, by high–temperature (high–TC) superconductors we understood that
discovered 35 years ago in Lax-1BxTiOy, Lax-1SrxCuOy, and YBa2Cu3O7-δ systems.
Later, the iron pnictide and chalogenide systems such as LaFeAsO1-xFx and
FeSe. Recently, the nickelates LaNiO2 and related compounds have been studied
intensively. A separate class is formed by the LaH10+x, for which the critical
temperature has reached 250 K or even higher value. The principal difference
between the hydrogen–rich and remaining system is that in the case of LaH10+x
the pairing of electrons seems to be caused by phonons, whereas in the
cuprates, nickelates, and iron pnictides the strong to moderate
interelectronic correlations play a decisive role. The aim of this brief
overview is to compare our theoretical results for the cuprates with the
principal experimental results in a consistent and quantitative way.

The structure of this brief review is as follows. In Sec. II we discuss
principal characteristics of the cuprates. In Sec. III we overview the
qualitative features of our theory, whereas in Sec. IV we provide explicit
examples of a quantitative comparison of our results with experimental data. A
brief outlook is deferred to Sec. V. This paper aims to specify and summarize
the most important results elaborated in a recent topical review [1].

##  II Principal characteristics of the cuprates

###  II.1 Structural and electronic specific features of the cuprates

The most striking structural property of the high–temperature superconducting
cuprates and pnictides is their quasi–two–dimensionality, composed in the
simplest situation of well separated CuO2 planes. This is the case for e.g.
La1-δSrδCuO4 or Bi2Sr1.6La0.4CuO6-δ mixed compounds. This simplifying
assumption induced series of studies of strictly two–dimensional models of
high-$T_{C}$ superconductivity, even though it is not exactly clear whether,
strictly speaking, a spatially homogeneous two–dimensional transition to
superconducting state is possible at nonzero temperature ($T>0$). The evidence
that such an ordering is supported by the results for single-plane of FeSe
[2].

The second most important feature is that the carriers in CuO2 plane are holes
in the doped Mott insulator. The situation is schematically depicted in Fig.
1. Note that only Cu2+ ions are shown for the sake clarity. Virtual hopping
processes, specified also there, lead to the antiferromagnetic kinetic
exchange representing the superexchange, whereas the real hopping processes
provide the charge transport of single carriers and their pairing in both
hole– and electron–doped cases situations. Note a gradual character of
transformation from antiferromagnetic Mott insulator to a strongly correlated
metal with doping. At this point, it is fair to say that so far it is not
clear whether this changeover from the Mott insulator to the strongly
correlated metal is a real quantum phase transition with an incipient quantum
critical point, blurred by the substitutional disorder (e.g. Sr for La),
taking place in the La–Sr–O insulating planes, sandwiching the periodic
arrangement of the CuO2 planes, where the actions goes.

Another striking feature is the circumstance that the CuO2 planes may be
represented originally by the atomic 3d${}_{{x}^{2}-{y}^{2}}$ states,
representing the highest positioned electron of nominally 3d9 shell of Cu2+
ion, hybridized with two 2px and 2py states of nominally O2- ions. The
situation is depicted schematically in Fig. 2 (left). This 3–orbital periodic
structure, arranged into a square lattice, leads to the three bands specified
in Fig. 2b; the tαβ parameters are the hopping integrals between the specified
orbitals.

Figure 1: Schematic representation of the particle dynamics in terms of
hopping processes (dashed arrows) in the Mott-insulating state (a) and the
strongly-correlated metal phase (b). Virtual hopping involves two consecutive
direct hopping processes back and forth and occurs in both cases, (a) and (b).
The direct hopping results in real motion of holes and occurs only in the
strongly-correlated metal phase (b). In the strong-correlation regime, the
direct hopping processes via doubly occupied configurations
$\ket{\uparrow\downarrow}$ are precluded. In the last case we speak about
extreme strong correlations. The arrows surrounding the hole (red circle) mark
possible real hoppings around it.

Now, if the CuO2 system is regarded as effectively single–band system composed
of 3d${}_{{x}^{2}-{y}^{2}}$ orbitals dressed with 2px,y orbitals, in which the
latter states play only a passive role [3], then such a single–band Mott
insulator is represented by a singly occupied set of Cu2+ ions. This is the
situation depicted in Fig. 1, where the $\uparrow$ and $\downarrow$ arrows
specify the spins of the ninth electrons per site, may be regarded as the
model situation of the Mott–Hubbard insulator. The doping $\delta\equiv 1-n$
represents then the average number of hole carriers counted per site. If
however, one takes the three-orbital 3d${}_{{x}^{2}-{y}^{2}}$–2px,y model,
then the corresponding Mott-Hubbard insulator (called in that case the
charge–transfer insulator) contains 5 electrons per Cu2+O${}^{2-}_{2}$ cluster
(two 2p electrons per oxygen and one electrons per copper). In that situation
the hole doping may be defined as $\delta\equiv 5-n$, where $n$ in both
situations is the number of electrons per fundamental unit, Cu or CuO2,
respectively.

Figure 2: Bare (without interaction) three–band structure in the tight binding
approximation. (a) Definition of the hopping parameters between the px , py ,
and d${}_{{x}^{2}-{y}^{2}}$ orbitals, with the sign convention for the
antibonding orbital structure. This structural unit forms a basis of three-
band model of the CuO2 plane in the cuprates. (b) The band structure of the
$d$-$p$ model with microscopic parameters: $t_{pd}\sim 1.13\text{ }eV$,
$t_{pp}\sim 0.49\text{ }eV$, and $\epsilon_{pd}\sim 3.57\text{ }eV$. The Fermi
energy is taken as the reference value and corresponds to the filling $n=5$
per Cu2+O${}_{2}^{2-}$ complex, corresponding to half–filled antibonding band.
This partially filled band is split off by about $\epsilon_{pd}$ from the
remaining filled bands and reflects a single–hybridized (bare) band, the
horizontal line marks the position of Fermi energy for $n=5$ electrons per
CuO2 unit (after [2]). In the strong correlation limit the antibonding band
(and the other two) is split into two Hubbard subbands.

The fourth principal assumption is that the original microscopic parameters
such as the hoppings $t_{\alpha\beta}$ or interaction strength do not vary
essentially in the whole doping range where superconductivity appear, i.e.,
for $0\lesssim\delta\lesssim 1/3$. Those bare parameters do vary from system
to system, but mainly due to interelectronic correlations which induce their
strong doping dependent renormalization. They complement the bare one–electron
structure. Effectively, one should regard the single–band model description as
that referring to the situation of the antibonding band (cf. Fig. 2b),
containing effectively $1-\delta$ electrons per copper. This point is to be
verified later.

The final structural feature of the system is the experimental observation
that the electronic properties in the normal state are those of practically
two–dimensional metal, with the resistivity in plane/across plane
$\rho_{\parallel}/\rho_{\perp}\sim 10^{5}$ in the optimal situation, and with
metallic/semiconducting behavior of $\rho_{\parallel}/\rho_{\perp}$,
respectively. On the contrary, the superconducting phase is three–dimensional.
This means that there is $d=2$ to $d=3$ dimensional changeover at the critical
temperature. In other words, the interplanar coherence appears in the
condensed state. This can be clearly shown when examining systematically
single– versus multi–planar systems critical temperature TC as a function of
the number of closely spaced planes [4].

###  II.2 Theoretical models of strongly correlated square planar structure of
the cuprates

In our group we have concentrated on studying two theoretical models: The
(extended) single–band Hubbard model under the acronym of t–U–J–V model (cf,
Appendix A), as well as on three–band 3d–2px,y model. In the latter situation
also its similarity to the one–band case under special circumstances has been
explored [3]. The first of them represents the most general single–band model
with short–range intersite interactions (and correlations), which reduces to
either $t$-$J$ or Hubbard model in proper limits. The three–band model, in
turn, allows for an explicit discussion of the role of oxygen in the particle
dynamics and ordering, particularly in the metallic state. We overview each of
them separately in the context of concrete results and compare them with
experiment.

##  III The method and its qualitative interpretation

###  III.1 The method: Single–band model

The most general single–band model of correlated electrons has been discussed
briefly in Appendix A. In this section we limit ourselves to the so–called
$t$-$J$-$U$-V model in the form [5]

$\displaystyle\hat{\mathcal{H}}=\sideset{}{{}^{\prime}}{\sum}_{ij\sigma}t_{ij}\hat{a}_{i\sigma}^{\dagger}\hat{a}_{j\sigma}+U\sum_{i}\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}+\sideset{}{{}^{\prime}}{\sum}_{ij}J_{ij}\hat{\mathbf{S}}_{i}\hat{\mathbf{S}}_{j}$
(1)
$\displaystyle+\frac{1}{2}\sideset{}{{}^{\prime}}{\sum}_{ij}\left(V_{ij}-\frac{1}{2}J_{ij}\right)\hat{n}_{i}\hat{n}_{j}.$

Figure 3: (a) Plot of the experimentally observed pseudogap (points) as
compared to theoretical results for the effective single–particle gap
$\Delta_{eff}$ obtained within the three–band model. The Hamiltonian
parameters are: $U_{d}=11\text{ }eV$, $U_{p}=4.1\text{ }eV$,
$\epsilon_{dp}=3.2\text{ }eV$, $t_{pp}=11\text{ }eV$. The critical doping
levels and the Mott–insulator boundary is also marked. (b) Relative correlated
$d$-wave gap component with intersite Coulomb interaction of magnitude
$V_{dd}=0.7\text{ }eV$. Experimental data sets 1 and 2 are taken from Ref.
[6]. For a brief discussion of the role of quantum fluctuations in bringing
the theoretical results to those obtained from experiment see Outlook.

The parameters and consecutive terms are defined and explained in Appendix A.
Hamiltonian (1) is used as a starting point for a further analysis and
solution fro many–particle states. In our comprehensive review [1] we selected
the approach based on a trial variational wave function and subsequently have
constructed a systematic diagrammatic expansion (DE–GWF) which in the lowest
order, provides renormalized mean–field theory (RMFT) in the form of
statistically consistent Gutzwiller approximation (SGA) [7]. In general, the
approach is based on selection of the ground–state many–particle wave function
$\ket{\psi_{G}}$ in the form

$\displaystyle\ket{\psi_{G}}\equiv\hat{P}\ket{\psi_{0}},$ (2)

where $\ket{\psi_{0}}$ represents an uncorrelated (single–particle) state, to
be defined later in the process of solving the model in a self-consistent
manner. The nontrivial projection operator $\hat{P}$ is given by [8]

$\displaystyle\hat{P}\equiv\prod_{i}\hat{P}_{i}=\prod_{i}\lambda_{i\Gamma}\ket{\Gamma}_{i}{}_{i}\bra{\Gamma}$
(3)

with the wave–function variational parameters
$\lambda_{i,\Gamma}\in\\{\lambda_{i,0},\lambda_{i,\uparrow},\lambda_{i,\downarrow},\lambda_{i,\uparrow\downarrow}\\}$,
corresponding to the local (lattice site i) states $\ket{\Phi}$,
$\ket{\uparrow}_{i}$, $\ket{\downarrow}_{i}$, and
$\ket{\uparrow\downarrow}_{i}$, respectively. The consecutive states represent
the empty, single occupied with spin quantum number $\uparrow$ and
$\downarrow$, and doubly–occupied states, all on site $i$. For such a choice
of the site representation, the $\lambda_{i,\Gamma}$ parameters weight the
relative probability amplitudes of local occupancies appearance for each site.
In the limit of large Coulomb repulsion ($U\gg W$, where W is the bare
bandwidth) the double occupancies are absent. Additionally, we consider here
translationally invariant paramagnetic state for which
$\lambda_{i\uparrow}=\lambda_{i\downarrow}=\lambda_{i}$.

The ground state energy is determined by minimizing the variational expression
for the ground state energy

$E_{G}\equiv\langle\mathcal{\hat{H}}\rangle_{G}=\frac{\langle\Psi_{G}|\mathcal{\hat{H}}|\Psi_{G}\rangle}{\langle\Psi_{G}|\Psi_{G}\rangle}=\frac{\langle\Psi_{0}|\hat{P}\mathcal{\hat{H}}\hat{P}|\Psi_{0}\rangle}{\langle\Psi_{0}|\hat{P}^{2}|\Psi_{0}\rangle},$
(4)

It turns out that by introducing the following additional ansatz for the
$\hat{P}_{i}$ operator [8]

$\displaystyle\hat{P}^{2}_{i}\equiv 1+xd^{HF}_{i},$ (5)

where $x$ is yet another variational parameter and by defining the quantities

$\displaystyle
d^{HF}_{i}\equiv\hat{n}^{HF}_{i\uparrow}\hat{n}^{HF}_{i\downarrow},\text{
}\hat{n}^{HF}_{i\sigma}\equiv\hat{n}_{i\sigma}-\braket{\hat{n}_{i\sigma}}\equiv\hat{n}_{i\sigma}-n_{0},$
(6)

with $n_{0}\equiv\braket{\psi_{0}}{\hat{n}_{i\sigma}}{\psi_{0}}$, we can
perform a systematic expansion of the functioanl (4) (for details see [1, 3])
and obtain explicitly the interesting us physical properties in the correlated
state which are determined thorough corresponding quantities in uncorrelated
state. Before detailed physical discussion we should mention the method of
defining the uncorrelated wave function $\psi_{0}$. Namely, it is determined
from another variational principle [9]

$\displaystyle\frac{\delta}{\delta\bra{\psi_{0}}}\\{\mathcal{F}-\lambda(\braket{\psi_{0}}{\psi_{0}}-1)\\}=0,$
(7)

where $\mathcal{F}\equiv\braket{\hat{\mathcal{H}}}_{G}$ expressed in terms of
uncorrelated correlation functions; here there two intersite functions

$\displaystyle
P_{ij}\equiv\braket{\hat{c}^{\dagger}_{i\sigma}\hat{c}_{j\sigma}},\text{
}S_{ij}\equiv\braket{\hat{c}^{\dagger}_{i\uparrow}\hat{c}^{\dagger}_{j\downarrow}}_{0},$
(8)

i.e., they represent averages of local hopping and pairing correlations in an
uncorrelated state. $\lambda$ is variational parameter introduced to ensure
that the wave function is normalized. The procedure of solving (7) is
equivalent to diagonalization of the effective Hamiltonian

$\displaystyle\mathcal{\hat{H}}^{eff}\equiv\sum_{ij\sigma}t_{ij}\hat{a}^{\dagger}_{i\sigma}\hat{a}_{j\sigma}+\sum_{ij}(\Delta_{ij}\hat{a}^{\dagger}_{i\uparrow}\hat{a}^{\dagger}_{j\downarrow}+\Delta^{*}\hat{a}_{i\downarrow}\hat{a}_{j\uparrow}),$
(9)

where

$\displaystyle t_{ij}^{eff}\equiv\frac{\delta\mathcal{F}}{\delta P_{ij}},$
(10)

and

$\displaystyle\Delta^{eff}_{ij}\equiv\frac{\delta\mathcal{F}}{\delta S_{ij}}.$
(11)

In effect, the determination of the uncorrelated properties reduces to the
diagonalization of the BCS–type Hamiltonian, and in turn, to that determining
the properties in the correlated state. As the averages in that state are
factorized in terms of uncorrelated $S_{ij}$ and $P_{ij}$, the latter
procedure completes the determination of $\ket{\psi_{G}}$ provided the
remaining variational parameters are also determined [1, 3].

###  III.2 Three-band model: A brief perspective

As said above, strictly speaking, the elementary structural unit in two
dimensions contains a single 3d${}_{{x}^{2}-{y}^{2}}$ orbital due to the ninth
electron of Cu2+ ions and two 2px and 2py orbitals, each filled with two
electrons in the parent (undoped) situation. Therefore, one has to formulate a
three–band model to see at least, what is its connection to the widely used
$t$-$J$-$U$-$V$ single–band models which should be regarded as a particular
case of the present one. For that purpose, one starts from the Hamiltonian

$\displaystyle\hat{\mathcal{H}}=\sum_{ij]^{\prime}\alpha\sigma}t^{pp}_{ij}\hat{p}_{i\alpha\sigma}^{\dagger}\hat{p}_{j\alpha\sigma}+\epsilon_{dp}\sum_{i\sigma}\hat{d}_{i\sigma}^{\dagger}\hat{d}_{i\sigma}$
$\displaystyle+\sideset{}{{}^{\prime}}{\sum}_{ij\alpha\sigma}t_{ij}^{pd}(\hat{d}^{\dagger}_{i\sigma}\hat{p}_{i\alpha\sigma}+\textrm{H.c.})$
(12)
$\displaystyle+U_{d}\sum_{i}\hat{n}_{di\uparrow}\hat{n}_{di\downarrow}+U_{p}\sum_{i}\hat{n}_{pi\alpha\uparrow}\hat{n}_{pi\alpha\downarrow}.$

.

Figure 4: (a) Experimental energy dispersion along the nodal direction for
$\mathrm{La_{1.9}Sr_{0.1}CuO_{4}}$ extracted from Refs. [10, 11]. The slopes
of solid lines are obtained theoretically from the effective Hamiltonian (red)
and first moment of the electron spectral function (red) for $\delta=0.1$. (b)
Doping-dependence of quasiparticle characteristic velocities above and below
the kink (green and blue squares, respectively). Corresponding green and blue
lines represent calculated effective- and correlated velocities calculated
using $\mathbf{k}$-DE-GWF method. The red line is the correlated velocity
multiplied by the calculated quasiparticle weight $Z_{\mathbf{k}}$. (c)
Calculated $Z_{\textbf{k}}$ as a function of doping (black points and lines),
compared with experimental data for
$\mathrm{Bi_{2}Sr_{2}CaCu_{2}O_{8+\mathit{x}}}$ (red points, extracted from
Ref. [12]). (d) Calculated Fermi wave vector along the nodal direction
compared with data for $\mathrm{La_{2-\mathit{x}}Sr_{\mathit{x}}CuO_{4}}$ for
the same parameter values as those used in fitting Figures (a)-(c). After
[13].

In that model version the bare $p$-$p$ hopping is assumed as nonzero only
between $nn$ 2p electrons, with $t_{pp}\sim 0.5\div 1\text{ }eV$,
$t_{pd}\simeq 1.1\div 1.3\text{ }eV$ is the single–particle hybridization
between 3d and 2p states and induces an effective $d$-$d$ hopping in the
higher order for the relevant antibonding states,
$\epsilon_{d}-\epsilon_{p}\simeq 3.5\text{ }eV$ is the so–called p
$\rightarrow$ d charge transfer energy, whereas the relevant intraatomic
$p$-$p$ and $d$-$d$ interactions have magnitudes $U_{pp}\simeq 4\div 5\text{
}eV$, and $U_{dd}\simeq 8\div 1\text{ }eV$. The other interactions term such
as, e.g., that $\sim U_{pd}$ are neglected, what is probably an oversimplified
feature of our model.

The question is to what extent results of the one–band model represented by
(1) and that starting from (III.2) are principally equivalent? A simple answer
to this question is provided by inspection of the Fig. 2b and noting that in
the doped systems the interesting regime is then for $\delta\lesssim 1/3$.
This, in effect, means that in the present version of the model the holes are
located in the antibonding band, since the p–d charge transfer gap
$\epsilon_{pd}=\epsilon_{d}-\epsilon_{p}$ is quite large on the scale of all
parameters except $U_{dd}$. This bare–band picture persists also upon
inclusion of interaction as the neglected $U_{pd}$ enlarges the charge
transfer gap and $U_{dd}$ does not reverse the trend, when $n_{d}\lesssim 1$.
The variational procedure presented in Sec. 3B is more involved [3] so it will
not be presented in this minireview. Perhaps, it is worth showing explicitly
the comparison of the doping dependence of the $d$-$d$ gap (cf. Fig. 2(a)) in
three–band model with that of single band model (cf. Fig. 2b); the consecutive
components $\Delta^{(i)}$ represent those between i-th neighbors (i-th
coordination sphere). The component $\Delta^{(2)}$ is absent for the
d–electrons, since the nearest neighboring $d$-$d$ correlations are strongly
antiferromagnetic, inducing the effective spin–triplet correlations between
the second neighbors. This exclusion does not appear for $d$-$p$ and $p$-$p$
pairing components; but they are of rather minor importance [3]. The two
figures in the panel illustrate thus to what extent two models (1– vs.
3–bands) may be regarded as equivalent .

###  III.3 The main qualitative features of the approach

First of all, the two energy scales appear in a natural way, i.e., that
described by $\ket{\psi_{G}}$ and $\ket{\psi_{0}}$, respectively, as
exemplified by the physical gap $\Delta_{G}$ and pseudogap $\Delta_{eff}$. The
question is whether those separate scales can be see in the actual correlated
state. To illustrate the two faces of the correlated fermionic liquid we have
compared first the doping dependence of the so–called pseudogap and the
superconducting gap, $\Delta_{eff}$ and $\Delta_{dd}$, respectively. Those
values have been compared with exemplary experimental results in Fig 3ab. The
amplitude $\Delta_{eff}$ is that obtained from the single–particle Hamiltonian
(III.2), whereas $\Delta_{dd}\equiv\Delta_{G}$ is that from solving the full
expression $\braket{\mathcal{H}}_{G}$. Both gaps have d–wave symmetry
$\Delta_{G,eff}(\textbf{k})=\Delta_{G,\text{eff}}(\cos k_{x}-\cos k_{y})$ from
Fig. 5ab. We see that the agreement of our theory (SGA) with experiment is
rather qualitative, as only the trend of the data is reproduced. However, we
believe that the inclusion of correlations (cf. Sec. V) may improve the
results exhibit in Fig. 5a essentially. Obviously, it is still to be carried
out in the future (see also Fig. 3)b).

Figure 5: Doping dependencies of the SC gap $Delta_{\text{eff}}(\textbf{k})$
at $\textbf{k}=(\pi,0)$ for different approximation schemes [7] 1–6 and for
$t^{\prime}/t=-0.27$ and $J/|t|=0.3$. Large filled circles: experimental data.
Two values of $t$ have been selected.

As the second test of the two energy scales we consider the single–electron
dispersion relation obtained from ARPES experiment. The exemplary comparison
of our modified approach [13] to experiment is shown in Fig. 5b. We see that
the comparison of the shifted by $\Delta k_{0}$ of the linear dispersion
relation for the correlated particles (the part below the kink) with respect
to that close to the Fermi energy. Note that $\Delta k\equiv k-k_{F}$ is the
wave vector measured with respect to the Fermi surface point ($k=k_{F}$) in
the nodal direction. In Fig. 4b we exhibit the spectral two Fermi velocities
$v^{eff}_{F,\text{low}}$ and $v^{corr}_{F}$, together with the renormalization
factor $Z_{\text{nodal}}$ in the latter case (cf. the corresponding dependence
in Fig. 4c). Remaining labeling on those curves is self-explanatory [13].

Figure 6: Doping dependence of Fermi velocity in the nodal [(0,0)
$\rightarrow$ ($\pi$,$\pi$)] direction. Experimental data are marked by
diamonds (after [7]) YBa2Cu3O7-δ (YBCO), squares (LSCO), and solid circles
(BSCCO).Two $t$ values have been selected.

The basic question to ask is whether such a division into effective Landau
quasiparticles and correlated particles of this quantum liquid (particles
above and below the kink) is physically feasible. Our interpretation is that
excitations the Fermi level (in the nodal direction) can be regarded as true
quasiparticles in the Landau sense, albeit renormalized differently, since our
starting interaction comprises all relevant itinerant electrons and is
short–range and strong in real space. On the contrary, the single–electron
excitations from the region deeper below the Fermi level (with energy
$\Delta\epsilon\equiv|\epsilon-\epsilon_{F}|\gtrsim 0.1\text{ }eV$) are
dressed with the full interaction, in which the Hubbard term plays predominant
role. Obviously, this division of a single quantum liquid of indistinguishable
quantum particles into two parts is qualitative in nature and signals (by the
kink’s existence) a crossover behavior from a liquid of diluted quasiparticles
to their truly correlated counterparts as one probes deeper into the Fermi
sea. Such an interpretation requires a further test as it squares well with
experiment (see also further evidence in the next Section). The division is
coded in the selection of the wave function in the form (2), which contains a
nonunitary projector $\hat{P}$, and is amplified by the fact that the starting
(uncorrelated) wave function $\ket{\psi_{0}}$ is also of nontrivial nature and
determined in a self–consistent manner that encompases also the states with
broken symmetry from start. We should note at the end that such a mixed
Fermi–non–Fermi liquid properties have been also observed in the transport
properties [14].

##  IV Detailed testing of the theory: Equilibrium properties

In this Section we discuss selected detailed characteristic of high
temperature superconductors obtained within our real–space pairing among all
itinerant electrons in our two–dimensional system.

###  IV.1 Inadequacy of the renormalized mean–field theory

Figure 7: Selected superconducting properties: (a) Kinetic energy gain $\Delta
E_{kin}$ vs. relative hole doping $\delta-\delta_{opt}$ ($\delta_{opt}$ is the
optimal doping). The microscopic parameters are $J=0.2|t|$, $U=22.6|t|$ (for
blue solid lines) and $J=0.2|t|$, $U=16|t|$ (for red solid lines); the
experimental points are taken from Ref. [5]. For comparison, the results
obtained with SGA method (gray dashed line) and those for the t-J model
($J=0.25|t|$) in DE–GWF approximation (dash–dotted line) are also included.
Note that only the $t$-$J$-$U$ model solution describes the data in a
quantitative manner. (b) correlated-gap magnitude $\Delta_{G}$ and (c) the
condensation energy $\Delta E_{C}=E^{SC}_{G}-E^{PM}_{G}$, both vs. $\delta$,
are also shown for the respective values of microscopic parameters and models.

Our work started with the analysis of the so–called renormalized mean–field
theory (RMFT), which has been very popular in the first decade after the
discovery of superconductivity in the nonstochiometric oxide La2-xBaxCuO4-δ
and YBa2Cu3O7-δ. The approach was originally based on an improved version of
the Gutzwiller approximation [15]. In our case, it take the form of
statistically consistent Gutzwiller approximation (SGA) [7]. In this
approximation the regime of doping, where the superconductivity qualitatively
as shown in Fig. 5ab. The presence of antiferromagnetism at low doping can be
reproduced qualitatively only after a careful selection of the detailed SGA
approximation scheme is carried out [16]. Furthermore, both the dependences of
the (correlated) superconducting gap (cf. Fig 5a) and particularly, of the
dispersion relation of the single–particle excitations, obtained from ARPES
(cf. Fig. 5b) are not reproduced correctly. Explicitly, as we can see from the
data included in Fig. 6b and on the basis of our later analysis based on the
full DE-GWF (cf. Fig. 4), the Fermi velocity is rather flat, whereas the
theoretical results shown in Fig. 6b exhibit Fermi-liquid type of relative
energy, diminishing steadily with decreasing doping. These results forced us
to look for a theory, in which the SGA (or RMFT) results can be corrected is
an essential way. In the next subsection we provide selected principal results
illustrating the usefulness of our DE-GWF approach.

###  IV.2 Additional results: Beyond mean field theory and comparison with
experiment

The most striking result to a theorist may be the fact, discovered
experimentally some time ago [17], is that the transition to the
superconducting state, particularly in the regime of low doping,
$\delta\lesssim 0.1$, takes place with the kinetic energy of the system
getting lowered by the transition from the paramagnetic to superconducting
phase. This is shown in Fig. 7a, where the results (squares with the error
marked) have been plotted against the relative doping
$\delta-\delta_{\text{opt}}$, where $\delta_{\text{opt}}$ is optimal doping.
Our theoretical curves require a more detailed explanation. Namely, the full
curves represent our DE–GWF solutions for two slightly different values of
parameters within $t$-$J$-$U$ model [3]. The other two (dashed and dot-dashed)
curves represent the SGA and $t$-$J$ model (beyond–SGA) solutions,
respectively. None of the latter two solutions reproduces the singular
behavior at low doping, at least for the type of detailed approach chosen.
Parenthetically, the fact that only the $t$-$J$-$U$ model, combined
additionally with the DE–GWF, reflects the data in a quantitative manner,
tells us that in order to reproduce fully them, one is forced to go beyond
either the Hubbard or $t$-$J$ model. In such a situation, we interpret the
simultaneous presence of both the Hubbard term with realistic values of $U\sim
8-10\text{ }eV$ and the kinetic exchange with its superexchange magnitude
$J\sim 0.1\text{ }eV$, as an implicit influence of the anionic 2px,y bands,
not included in the standard one–band model, and producing the exchange
interaction of desired magnitude, while keeping the Hubbard $U$ in the
realistic range at the same time.

In Figs. 7(b) and (c) we show the correlated gap magnitude of the d-wave
solution and the condensation energy, respectively (the curve labelling and
their meaning is the same as that in Fig. 7a). Note that $\Delta_{G}\sim
0.03-0.05=15\text{ }meV\sim 160\text{ }K$ which is of the order of
experimental value of $T_{{C}}$, but is substantially higher. This last fact
is understandable as we do not account for thermodynamic fluctuation. Also,
the condensation energy, i.e., the difference between the ground–state
energies in normal and SC states is of the same magnitude and is strongly, but
systematically, decreasing with increasing $\delta$. Comparing Figs. 7(a) and
(c) we see that surprising lowering with diminishing $\delta$ is related to
the corresponding kinetic–energy decrease. The lowering of $\Delta_{G}$ with
$\delta\rightarrow 0$ is caused by the Mott–Hubbard localization effects
(renormalization of $|t|$) so that the two quantities do behave differently
near that limit. Such difference in behavior may be the sign of the quantum
spin–liquid effects, which are interrupted by the carrier localization. For
detailed discussion of phase diagram and associated with it crossover from
non–BCS to BCS–like see [3].

Figure 8: The phase diagram comprising various charge–density–wave states: (a)
theory and (b) experiment [3, 18]. For detailed discussion of various
order–parameter components see [3]. Note that the onset of pair–density wave
(PDW) induces also a small s–wave type of ordering in the system with the
primary d–wave SC ordering. Pure d–wave superconducting phase appears only at
and above the optimal doping, as observed.

In the last decade, the presence of the charge–order presence has been
intensely discussed, also in the context of the appearance of hidden charge
density–wave quantum critical point at the optimal doping [17] . Leaving aside
a detailed discussion, we have analyzed the effect of finite–range
correlations within our DE–GWF method [3] on the appearance of the CDW–type
state also with possible pair–density–wave presence have those states into the
phase diagram. We plot in Fig. 8a the theoretical results and have compared
them with experimental data [19, 3]. Note the qualitative agreement between
the two. In theory the most remarkable is the charge–splitting analog to the
Fulde–Ferrell state in the split electronic structure which seems to reflect
the experimental shape of the phase diagram. In passing, one can note the
astonishing richness of the phases for this model two–dimensional structure.
The relation of those features to the persistence of the van–Hove singularity
in the correlated state, should perhaps be discussed in more detail.

To illustrate further the relevance of our results, we have calculated for the
(approximately) same values of the microscopic parameters the selected
single–particle characteristics in the correlated state and within the
three–band model [3]. They are quite similar to those obtained within the
single–band model [6]. Explicitly, in Fig. 9 we display the doping dependence
of the Fermi velocity (a), Fermi wave vector (b) and effective mass (c). What
is surprising is a rather weak $\delta$ dependence in throughout the metallic
phase. The dotted line represents out theoretical results, in the case (c) for
two systems: La–Sr–Cu–O (LSCO) and Y–Ba–Cu–O (YBCO), respectively. Therefore,
it is tempting to say that a nonmonotonic dependence of the critical
temperature (TC) or the correlated superconducting gap magnitude $\Delta_{G}$
is induced mainly by the competing character of the kinetic, exchange, and
intraatomic Hubbard interactions. An analogical situation arises in the
systems near the Mott–Hubbard insulator–metal transition [20].

Figure 9: The basic characteristics calculated for the three-band model. From
top to bottom: Fermi velocity $v_{F}$ , Fermi wave–vector $k_{F}$ , and
effective mass enhancement $m_{eff}/m_{e}$ ; all as a function of hole doping
$\delta$. The parameters are: $t_{pd}=1\text{}eV$, $t_{pp}=0.4\text{}eV$,
$\epsilon_{pd}=3.2\text{}eV$, $U_{d}=11\text{}eV$, and $U_{p}=4.1\text{}eV$.
Note a quite smooth $\delta$ dependence of all the single–electron parameters.
Figure 10: Imaginary parts of transverse dynamical spin susceptibility for
obtained within one-orbital Hubbard model with nearest- and next-nearest-
neighbor hopping integrals included, and comparison with experiment for
$\mathrm{La_{2-\delta}Sr_{\delta}CuO_{4}}$ (LSCO). The model parameters are
$t=-0.34\,\mathrm{eV}$, $U=7|t|$. Panels (a) and (b) represent magnetic
response at half filling ($n=1$) in the antiferromagnetic state, obtained
within $\mathrm{SGA}_{x}$+$1/\mathcal{N}_{f}$ and RPA, respectively. The
spectra are similar and both of them match neutron scattering data for LSCO
(red circles). The dashed lines are the paramagnon energies obtained from
theoretical intensities using damped harmonic oscillator model. Panels (c) and
(d) result from the same analysis, but for hole-doped system ($n=0.88$) in the
paramagnetic state. Here, the differences are qualitative: the
$\mathrm{SGA}_{x}$+$1/\mathcal{N}_{f}$ method yields propagating magnetic
excitations along $\Gamma$-$M$ line, whereas within RPA one obtains overdamped
dynamics (see dashed curves and was discussion in the text). The agreement of
the SGA+$1/N$ with RIXS data (diamonds) is semi-quantitative. Panel (e) shows
$\mathrm{SGA}_{x}$+$1/\mathcal{N}_{f}$ results in the antiferromagnetic phase
at lower temperature. In (f)-(g) we compare the theoretical RPA and
$\mathrm{SGA}_{x}$+$1/\mathcal{N}_{f}$ paramagnon dispersion with experiment.
Adapted from Ref. [21]

At the end, one should note that a discussion of the onset on nemacity appears
also in the systems discussed here and discussed within DE-GWF [21]. All in
all, these results demonstrate the usefulness and effectiveness of the DE–GWF
method which represents a systematic approach beyond the mean–field type
approach for these strong correlated systems. The whole approach bases on
finite–U but large ($U\gtrsim W$) combined with strong superexchange
interactions.

##  V Extension: Paramagnons and plasmons dynamic excitations

So far, the whole DE–GWF analysis was based on taking into account, the static
intersite correlations of increased range [1], starting from SGA. We have
extended this analysis to collective dynamic excitations (paramagnons and
plasmons) by starting again from SGA and including long–range quantum
fluctuations in the lowest order within $1/N$ expansion [21, 22, 23, 24]. Here
we summarize briefly only the results for spin fluctuations spectrum in the
Gaussian approximation. The results are summarized in the panel composing Fig.
10. The theoretical results marked by the color scale (on the right of the
figures) and by the broken curves in figures (a)-(c). The curves in figures
(f) and (g) are compared with the theoretical results explicitly with
experiment, as well as show the differences for the doping $\delta=1-n=0.12$
between those obtained in random phase–approximation (RPA) and those obtained
in SGA (i.e., without correlations included). Only the full theory SGA+1/N
compares excellently with experiment. A similar theory can be formulated for
the plasmon excitations [21, 22, 23, 24] (see also the relevant contribution
to this volume [25]). One should mention that in describing the exhibited
paramagnon excitation characteristics, a standard damped oscillator
representation of the theoretical results was involved. This approximate
(Lorentzian) representation of the excitation spectrum should be considered
carefully, and perhaps, a more general approach is required. We should see a
progress along this line in the near future.

##  VI Outlook

We have overviewed here selected basic characteristics of high temperature
superconducting cuprates. This paper summarizes some of the main results
elaborated in detail in a comprehensive review [1]. The principal results and
their favorable (semi)quantitative comparison with experiment support the
fundamental concept of strong correlations combined with superexchange
(kinetic exchange in one–band version of the theory) as the mechanism of the
spin–singlet $d$-wave pairing in the cuprates. As we showed [1] and also here,
it is indispensable to formulate the theory beyond any version of the
(renormalized) mean field theory. What is still lacking is the incorporation
of the quantum spin and charge fluctuations in the single–particle description
of the normal–state properties to reproduce (or correct) the properties such
as the linear electrical resistivity or pseudogap presence, to obtain a more
complete quantitative picture. We should be able to see a progress along these
lines in the near future. Also, the form of single–band starting Hamiltonian
relation to its more–general three-band formulation with an explicit inclusion
of the superexchange in the metallic phase should be reanalyzed carefully.
Finally, the role of the third dimension is still to be incorporated on our
theory to see explicitly the correction (if any) of the apical (interplane)
oxygen may have in formation of the 3d superconducting state.

Parenthetically, this article concludes the series of the minireviews
published in the present journal over the years in this journal [26, 27]

## Acknowledgement

The work reported in this paper was financially supported by the grants OPUS
Nos. UMO–2018/29/B
/ST3/02646 and No. UMO–2021/41/B/ST3/04070. The cooperaton and multiple
discussions of the results wit Drs. Maciek Fidrysiak, Michał Zegrodnik, and
Andrzej Biborski have been the most important factor in making this project so
fruitful. I thank also to my Ph.D. student Maciek Hendzel for technical help
during the writing of this report.

## Appendix A: From classical Coulomb repulsion to extended Hubbard model

The classical repulsive Coulomb interaction between two charges is long–range,
changing with their mutual distance $|\textbf{r}_{i}-\textbf{r}_{j}|$ as

$\displaystyle V_{12}\equiv
V(\textbf{r}_{i}-\textbf{r}_{j})=\frac{1}{\kappa}\frac{q_{1}q_{2}}{|\textbf{r}_{i}-\textbf{r}_{j}|}.$
(13)

In the case of continuous charge densities $n(\textbf{r}_{i})$ and
$n(\textbf{r}_{j})$ it takes the static Lenard-Wiechert form

$\displaystyle V_{12}=\frac{e^{2}}{\kappa}\int
d^{3}\textbf{r}d^{3}\textbf{r}^{\prime}\frac{n(\textbf{r})n(\textbf{r}^{\prime})}{|\textbf{r}-\textbf{r}^{\prime}|}.$
(14)

For $n_{i}(\textbf{r})=\delta(\textbf{r}-\textbf{r}_{i})$ (14) reduces to
(13). In turn, in wave mechanics the interaction between two charges is then

$\displaystyle V_{12}=\frac{e^{2}}{\kappa}\int
d^{3}\textbf{r}d^{3}\textbf{r}^{\prime}\frac{|\phi_{1}(\textbf{r})|^{2}|\phi_{2}(\textbf{r}^{\prime})|^{2}}{|\textbf{r}-\textbf{r}^{\prime}|},$
(15)

where now $|\phi_{i}(\textbf{r})|^{2}$ is the probability density for i-th
particle. Finally, in quantum field theory the static interaction for
indistinguishable particles is of the following operator form

$\displaystyle\hat{V}=\frac{1}{2}\sum_{ijkl}\sum_{\sigma\sigma^{\prime}}V_{ijkl}\hat{a}_{i\sigma}^{\dagger}\hat{a}_{j\sigma^{\prime}}^{\dagger}\hat{a}_{l\sigma^{\prime}}\hat{a}_{k\sigma}$
(16)

with

$\displaystyle V_{ijkl}=\int
d^{3}\textbf{r}d^{3}\textbf{r}^{\prime}\phi^{*}_{i\sigma}(\textbf{r})\phi^{*}_{j}(\textbf{r}^{\prime})\frac{e^{2}}{\kappa|\textbf{r}-\textbf{r}^{\prime}|}\phi_{k}(\textbf{r})\phi_{l}(\textbf{r}^{\prime}).$
(17)

Indices $(i,j,k,l)$ run over all possible single–particle states
${\phi_{i\sigma}(\textbf{r})}_{i=1,2,...,N}$ and $(\sigma,\sigma^{\prime})$
are spin quantum numbers for particular fermions characterized by $(i,j,k,l)$.
We see that in (17) the probability densities $|\phi_{i}(\textbf{r})|^{2}$ and
$|\phi_{j}(\textbf{r}^{\prime})|^{2}$ are replaced by quantities
$\phi_{i}^{*}(\textbf{r})\phi_{j}(\textbf{r}^{\prime})$ and
$\phi_{k}(\textbf{r})\phi_{l}(\textbf{r}^{\prime})$, respectively. We may say
that they express roughly the overlap functions, but strictly speaking the
states $\phi_{i}(\textbf{r})$ and $\phi_{j}(\textbf{r})$ are usually
orthogonal, i.e. $\int
d^{3}\textbf{r}\phi^{*}_{i}(\textbf{r})\phi^{*}_{j}(\textbf{r})=\delta_{ij}$.

The interesting us question is what happens if the wave functions
${\phi_{i}(\textbf{r})}$ are close to their atomic correspondants. In that
limit $(i,j,k,l)$ when selected as parent atomic–state site positions are
sufficiently far from each other that the largest contribution is to (17)
comes from the term $i=j=k=l$ as in that case

$\displaystyle V_{iiii}=\frac{e^{2}}{k}\int
d^{3}\textbf{r}d^{3}\textbf{r}^{\prime}\frac{|\phi_{i}(\textbf{r})|^{2}|\phi_{j}(\textbf{r}^{\prime})|^{2}}{|\textbf{r}-\textbf{r}^{\prime}|},$
(18)

i.e., has the (15) from (effectively, also (14) form). This is the reason why
we call this term a director Coulomb interaction term. Then, (16) in
second–quantization reduces to the Hubbard interaction term if we take only
(18) out of all the terms appearing in (17), i.e.,

$\displaystyle\hat{V}_{ii}\equiv\frac{1}{2}\sum_{i\sigma}V_{iiii}\hat{n}_{i\sigma}\hat{n}_{i\bar{\sigma}}\equiv
U\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}.$ (19)

In the other words, this term predominates over all remaining terms if the
overlap functions $\phi^{*}_{i}(\textbf{r})\phi_{j}(\textbf{r}^{\prime})$ are
the only relevant quantities for $i=j$, i.e., the neighboring atomic states
$\phi_{i}(\textbf{r})$ and $\phi_{j}(\textbf{r})$ are well separated. This
assumption is the fundamental concept validating the Hubbard model which
represents a particular limit of (16), namely

$\displaystyle\hat{\mathcal{H}}=\sum_{ij\sigma}t_{ij}\hat{a}^{\dagger}_{i\sigma}\hat{a}_{j\sigma}+U\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}.$
(20)

This (still unsolved) model applies to so many physical systems, albeit often
only semiquantitatively.

In the present analysis important are also nearest–neighbor Coulomb
interactions. In that case, we have for two–state (two–site) terms namely

$\displaystyle\hat{V}=\frac{1}{2}{\sum_{ij}}^{\prime}K_{ij}\hat{n}_{i}\hat{n}_{j}-\frac{1}{2}{\sum_{ij}}^{\prime}J^{H}_{ij}(\hat{\textbf{S}}_{i}\hat{\textbf{S}}_{j}-\frac{1}{4}\hat{n}_{i}\hat{n}_{j})$
$\displaystyle+\frac{1}{2}{\sum_{ij\sigma}}^{\prime}{V_{ij}}^{\prime}(\hat{n}_{i\sigma}+\hat{n}_{j\sigma})(\hat{a}^{\dagger}_{i\bar{\sigma}}\hat{a}_{j\bar{\sigma}}+\hat{a}^{\dagger}_{j\bar{\sigma}}\hat{a}_{i\bar{\sigma}})$
(21)
$\displaystyle+{\sum_{ij}}^{\prime}{J}^{\prime}_{ij}(\hat{a}^{\dagger}_{i\uparrow}\hat{a}^{\dagger}_{i\downarrow}\hat{a}_{j\downarrow}\hat{a}_{j\uparrow}+\text{H.c.}),$

where the first term is the direct Coulomb intersite term, the second
represents direct (Heisenberg) exchange interaction, the third the so–called
correlated hopping, and the last pair hopping term (for details see [1]). The
first two terms, when added to (20) result in an extended Hubbard model. In
the second term we include in the exchange (second) term containing also the
effective kinetic exchange interaction. Hence, $J_{ij}$ becomes negative. In
effect, by redefining constants $J_{ij}\equiv J^{H}_{ij}-J^{\text{kex}}$,
$V_{ij}=K_{ij}/2+J_{ij}/4$ we obtain t–U–J–V model in the form

$\displaystyle\hat{\mathcal{H}}=\sideset{}{{}^{\prime}}{\sum}_{i\sigma}t_{ij}\hat{a}_{i\sigma}^{\dagger}\hat{a}_{i\sigma}+U\sum_{i}\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}+\sideset{}{{}^{\prime}}{\sum}_{ij}J_{ij}\hat{\mathbf{S}}_{i}\hat{\mathbf{S}}_{j}$
(22)
$\displaystyle+\frac{1}{2}\sideset{}{{}^{\prime}}{\sum}_{ij}V_{ij}\hat{n}_{i}\hat{n}_{j}.$

Results obtained with this model and its particular versions are discussed in
detail in the main text.

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# Shared Affordance-awareness via Augmented Reality
for Proactive Assistance in Human-robot Collaboration

Drake Moore
<EMAIL_ADDRESS>Mark Zolotas
<EMAIL_ADDRESS>Taşkın Padır
<EMAIL_ADDRESS>
###### Abstract

Enabling humans and robots to collaborate effectively requires purposeful
communication and an understanding of each other’s affordances. Prior work in
human-robot collaboration has incorporated knowledge of human affordances,
i.e., their action possibilities in the current context, into autonomous robot
decision-making. This “affordance-awareness” is especially promising for
service robots that need to know when and how to assist a person that cannot
independently complete a task. However, robots still fall short in performing
many common tasks autonomously. In this work-in-progress paper, we propose an
augmented reality (AR) framework that bridges the gap in an assistive robot’s
capabilities by actively engaging with a human through a shared affordance-
awareness representation. Leveraging the different perspectives from a human
wearing an AR headset and a robot’s equipped sensors, we can build a
perceptual representation of the shared environment and model regions of
respective agent affordances. The AR interface can also allow both agents to
communicate affordances with one another, as well as prompt for assistance
when attempting to perform an action outside their affordance region. This
paper presents the main components of the proposed framework and discusses its
potential through a domestic cleaning task experiment.

##  1 Introduction

Human-robot collaboration (HRC) is increasingly prevalent in industrial,
commercial, and household settings. Nevertheless, HRC remains a naturally
complex problem when robots must transition into environments with an
unpredictable structure [1], or when assisting a person with limited physical
capabilities due to disability [9]. In such scenarios, the robot and human
agents may encounter areas where they are unable to operate. Whenever an agent
is unable to access a desired object or area during a collaborative task, it
becomes vital to successful task completion that this state of deadlock is
made transparent to both agents [38, 16]. As a result, each agent in the HRC
must not only have knowledge of their own limitations, but should also have
sufficient information about their partner’s capabilities in order to
proactively request assistance if necessary.

Figure 1: Illustration of human and robot affordances in a collaborative
cleaning task. The human’s affordable area begins as a small ellipsoid
centered around their chest, extending to their hands. As the person grasps
and places an object into the allocated bin, their affordable area grows to
include novel areas they interacted with.

A prominent means of accurately modeling and conveying agent capabilities
within their environment is to represent these capabilities as object
affordances [13, 21, 18]. Defining object affordances as the possible actions
an agent can perform on an object [12, 18], recent works in HRC have utilized
knowledge of these affordances to inform the autonomous robot’s responsive
behavior [21, 30, 35]. However, many of these works have modeled either the
human or robot’s affordances, before then uni-directionally transmitting this
information to the other agent. Bi-directional communication of agent
intentions, affordances, and spatial perspective continues to be one of the
most pressing modern challenges for HRC [16].

In this paper, we propose an augmented reality (AR) framework to establish
“affordance-awareness” in both agents through a shared spatial understanding
of the collaborative task environment. Using RGB-D sensors available to the AR
headset and robot, the environment is perceived from both agent perspectives.
Object affordances are then computed in terms of reachability via agent-
specific models, where an object in reach is assumed affordable by the agent.
We model robot affordances as manipulability ellipsoids that are pre-computed
from the robot’s known kinematics and a previously mapped voxel grid of the
environment [42, 45, 46]. Human affordances are instead generated based on
kinematic models of human reachability [24, 17], and expressed as a binary
voxel grid. These individual agent perspectives and affordances produce a
shared spatial representation of affordable areas. Moreover, the AR interface
offers the human and robot a means of relaying information on their respective
affordance regions to one another.

Adopting this terminology for affordances, the remainder of this paper
introduces the main components of the proposed framework and grounds its
applicability in a domestic cleaning task (see Fig. 1). Under this task setup,
we demonstrate how each agent can reason over generated affordances to
proactively communicate intent, request assistance, and perform collaborative
behaviors.

##  2 Related Work

A plethora of research has investigated the application of AR technologies as
a communication medium in HRC. For instance, multiple studies have used AR
headsets to visualize the robot’s internal model and planned trajectories,
thereby improving human decision-making during collaboration [48, 6, 39].
Leveraging dual perspectives provided by an AR headset and the robot’s vision
sensors has also been used to build comprehensive spatial maps of a shared
environment to guide agent exploration [36]. AR interfaces in HRC have
additionally been explored to communicate robot intentions for the purpose of
improving human safety and trust [5, 43, 41]. These previous works exhibit
uni-directional communication of the robot’s state, which benefits solely the
human’s reasoning.

Bi-directional recognition of internal human and robot states is an auspicious
trend in the field of HRC [16], for which AR headsets hold great promise as a
bridge of communication [38]. Recent work has investigated methods of
portraying a robot’s sensory state to human partners and then enabling the
human user to interactively edit the robot’s resulting behavior [7, 8]. Over a
bi-directional communication channel, robots may also request assistance from
human users via the AR interface [31]. In these bi-directional communication
setups, the role of the human and robot agents becomes even more relevant,
specifically in terms of whether they are passive or active. Bi-directional
communication channels where both the human and robot could actively perceive
and manipulate holograms have been shown to yield better task performance
[34].

Another essential element of AR headset interfaces relates to how information
on agent capabilities, e.g., represented as affordances, is presented [48].
Affordances have been modeled in various forms to enhance the quality of HRC,
such as object grasp positions, spatial usability, object usage, or
reachability [21, 18, 32]. Object affordances can be modeled for both human
and robot manipulation to encode physical constraints in a 2-D semantic map
[11]. Alternatively, the number of agent visits to spatial areas and the type
of activity performed at that occupancy cell can be used to encode an
affordance representation of an environment [25, 2]. Affordance-aware AR user
interfaces for HRC have also been developed to present humans with a semantic
understanding of a robot’s atomic control actions, but no human-to-robot
communication was established [35].

While prior work has achieved effective bi-directional communication in HRC,
we posit that the advantages obtained from using AR headset interfaces have
not yet been fully leveraged. Our major insight is that the shared knowledge
of each agent’s affordances will prove more effective than a one-way
communication channel. To achieve shared affordance-awareness, we introduce an
AR framework that constructs an environmental representation of affordances
from the visual perspectives and estimated intentions of both agents in the
HRC. A key benefit of this bi-directional information flow is that the human
and robot can proactively request assistance from their partner whenever they
cannot afford an action. In this paper, the term affordances will from hereon
refer to the ability for an agent to reach and act upon given areas and
objects within the environment.

##  3 Shared Affordance-awareness Framework

Figure 2: Key components of the augmented reality (AR) framework for
affordance-awareness in human-robot collaboration. Perceptual information on
objects is obtained through an AR headset and a robot’s sensors to create a
shared spatial map. The affordable areas of the human and robot can then be
delineated from agent actions on this shared environment (denoted by red and
green circles). Modeled affordances, along with a bi-directional communication
channel of human and robot intent, will inform both agent’s decision-making.
The human and robot can also query each other for assistance (blue line) when
objects fall outside their respective affordable areas.

In this section, we describe a shared affordance-aware framework for HRC based
on an AR headset interface (see Fig. 2 for the architecture overview). The
framework combines both the action and perception spaces of the human and
robot. Therefore, the human and robot agents are now capable of querying their
partner for assistance whenever objects lie outside their own affordable
areas. The overall architecture will target the HoloLens 2 headset and a
mobile manipulator robot, built atop the ROS 2 middleware [28].

###  3.1 Intent Communication

In a collaborative task, human and robot agents must be able to communicate
and understand each other’s intent. There are many ways of representing agent
intent in HRC [26, 47], but for this work we describe intent as a target
object in 3-D Cartesian space that an agent $i$ plans to interact with,
$\mathbf{o}_{i}\in\mathbb{R}^{3}$. We also only consider a single human and
robot agent in the collaboration, ergo $i=\\{H,R\\}$.

We plan to adopt a similar approach to prior work in communicating robot
intent by displaying planned robot trajectories and safety constraints via the
user’s AR headset [43, 41]. From the egocentric AR headset’s perspective of
the human user’s hands and eye gaze, we can also predict hand and gaze
trajectories to estimate human intent [15, 40]. By presenting the robot with
an estimate of the human’s grasp plan, the robot can preemptively make
adjustments to continue performing safe and collaborative behavior [23].
Furthermore, a human can act around the robot’s planned behavior by relying on
the AR interface’s feedback.

###  3.2 Spatial Computing

Spatial computing is the ability for devices to track key features in an
environment and represent them digitally [8]. Azure Spatial Anchors (ASA) is a
mixed reality toolkit for the localization of egocentric devices. This service
can be run on both the HoloLens and a mobile robot via the open-source ROS
package111https://github.com/microsoft/azure_spatial_anchors_ros. Once spatial
anchors are determined and shared between the AR headset and robot, a general
world frame can be defined for the environment. By simultaneously running ASA
on the AR headset and mobile robot, the two sensory sources can be co-
localized in a common environment, thus establishing a shared world frame to
spatially represent environment features.

In addition to using ASA, we will employ popular object detection and tracking
methods to find the locations of objects pertaining to the collaborative task
[44]. For our task, object positions in 3-D space are estimated using the
center of the object’s bounding box in the image space, coupled with its
median filtered depth reading. All object positions can be localized in the
shared world frame, which is accessible to both the robot and human.

###  3.3 Affordance Modeling

Modeling affordances is an active multidisciplinary area of research with a
diverse array of computational methods [18]. In this work, we will simplify
the problem to treat affordances as spatial regions within an environment,
where these regions are delineated by the agent’s reachability of objects.
Every object in the environment that can be interacted with is assumed
“affordable” by both the human and robot, such that the affordances,
$\mathcal{A}_{i}$, of an agent $i$ do not need to explicitly model action-
object relations [21, 18]. Instead, $\mathcal{A}_{i}$ can be represented as a
voxel grid of reachable states estimated using the depth sensors of both
agents.

For human affordances, $\mathcal{A}_{H}$, the voxel grid must capture areas in
the environment where a human can operate [34, 20]. In an assistive robotics
setting, human agents will often be unwilling or unable to perform actions
outside of a given region. As a result, we begin by modeling the entire
environment as non-affordable to the human. Throughout the task, we will
continuously update the human affordances set, $\mathcal{A}_{H}$, with new
voxels based on the kinematic reachability of human arms, as observed from
both the user’s and robot’s perspectives [24, 37, 10].

To model robot affordances, $\mathcal{A}_{R}$, we will adopt existing methods
of reasoning over a robot’s kinematic constraints given a known spatial map of
the environment. These methods represent the capacity for a mobile manipulator
to reach and interact with the environment by storing the robot’s
manipulability ellipsoids at every pose in the discretized workspace [45, 46].
We pre-compute these manipulability ellipsoids for the entire environment in
an offline manner, resulting in a voxelized representation of the robot’s
affordable workspace [42]. While this provides an accurate estimate of the
robot’s affordable area, we still perform collision-free motion planning for
each target object to ensure the system is robust to environment changes.

###  3.4 Proactive Assistance Communication

By sharing representations of affordances and intentions to both agents in the
HRC, we present a framework that best informs the collaborative agent’s
decision-making. For example, the robot is given its own affordances, the
human’s affordances, the target object location, and the human’s intent, which
in turn can be used to guide the autonomous robot motion planning (see
Algorithm 1). A core benefit of using AR headsets in this framework is that
both agents are able to request assistance from their partner via verbal
commands or visualizations in the interface. Note how in Algorithm 1, the
robot queries for help from the human agent, e.g., as a graphical cue like a
directional arrow [48], whenever there is no collision-free trajectory,
$\tau_{R}$, towards its target object. By enabling this bi-directional
communication channel, human and robot agents can reason about the shared
state of the environment and proactively query each other for assistance
whenever necessary.

Input: Robot and human target objects, $\mathbf{o}_{R},\mathbf{o}_{H}$; Human
affordable area, $\mathcal{A}_{H}$; Detected objects, $\mathcal{O}$;

Output: Collaborative robot plan;

Initialize: Collision-free robot trajectory, $\tau_{R}$, towards target object
$\mathbf{o}_{R}\rightarrow\tau_{R}$;

if $\tau_{R}$ found and $\mathbf{o}_{R}\not=\mathbf{o}_{H}$ then

return $\tau_{R}$

else if $\tau_{R}$ found and $\mathbf{o}_{R}==\mathbf{o}_{H}$ then

re-plan using new $\mathbf{o}_{R}$ from $\mathcal{O}$

else if $\tau_{R}$ not found and $\mathbf{o}_{R}\in\mathcal{A}_{H}$ then

query human to move $\mathbf{o}_{R}$ into robot affordable area

else if $\tau_{R}$ not found and $\mathbf{o}_{R}\notin\mathcal{A}_{H}$ then

query human to see if $\mathbf{o}_{R}$ is reachable

end if

Algorithm 1 Shared affordance-awareness for collaborative mobile robot
manipulation of objects

##  4 Proposed Experiment

To evaluate the efficacy of the proposed affordance-aware framework, we
consider an HRC experiment involving a domestic cleaning task. In this task
setting, various objects from the YCB dataset [4] will be spread out among the
floor, tables, and shelves (Fig. 1). The task will require agents to collect
and deposit these objects into one of three bins located around the
environment, depending on the object’s category: food items, kitchen items,
and household tools. The varied arrangement of object starting locations and
categories presents notable difficulties for an individual mobile manipulator,
or a person with disabilities attempting to accomplish the task independently.

In contrast with no communication and uni-directional robot-to-human
communication, we hypothesize that our shared affordance-aware framework will:

  * •

H1: Demonstrate increased task efficiency, as measured by task completion
time, intent switching frequency, and agent idle time.

  * •

H2: Result in reduced mental workload for human participants, estimated by
NASA-TLX [14] and quantitative metrics based on eye-related measures, e.g.,
pupil diameter [33].

  * •

H3: Enhance usability ratings, as indicated by the System Usability Scale
(SUS) [3].

###  4.1 Experimental Setup

In our domestic cleaning task, a between-subjects study will be conducted with
one of three experiment modes: non-communicative, robot-to-human communication
only, or shared affordance-aware communication. In the non-communicative mode,
the human and robot will act as independent agents with no means of sharing
intent or affordances. In the second configuration, only the robot agent’s
intent and affordances will be displayed in the AR interface, as in previous
work [35]. In the final experiment mode, our framework will communicate
intent, affordances, and allow agents to query one another for assistance.

###  4.2 Performance Metrics

We will evaluate the performance of the collaborative human-robot team through
subjective user responses and quantitative metrics. Using NASA-TLX [14] and
SUS [3] scales, we will acquire participant opinions on our system’s ease of
use, helpfulness, perceived workload, as well as overall user experience. By
equipping the AR headset with a Pupil Labs eye tracker add-on [19], we will
record gaze fixation and pupil dilation signals during the task, hence
providing suitable signifiers of the human agent’s workload [27, 22, 29].
Lastly, we evaluate team efficiency through agent idle time (duration where an
agent has no object of intent), frequency of intent switching, as well as
time-to-completion. These metrics will provide a comprehensive overview of our
framework’s ability to facilitate effective HRC in a real-world collaborative
task.

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