††thanks: The authors would like to thank for financial support from CINELDI –
the Centre for Intelligent Electricity Distribution, Research Council of
Norway project no. 257626/E20. Author 2 thanks Kuwait University for funding.
This paper was not presented at any IFAC meeting. Corresponding author R.R.
Bitmead. Tel. +1 858 822 3477.

# Simultaneous input & state estimation, singular filtering and stability

Mohammad Ali Abooshahab<EMAIL_ADDRESS>Mohammed M.J. Alyaseen
<EMAIL_ADDRESS>Robert R. Bitmead<EMAIL_ADDRESS>Morten Hovd
<EMAIL_ADDRESS>Department of Engineering Cybernetics, Norwegian
University of Science & Technology, 7491 Trondheim, Norway.  Department of
Mechanical & Aerospace Engineering, University of California, San Diego, 9500
Gilman Drive, MS-0411, La Jolla CA 92093-0411, USA.

###### Abstract

Input estimation is a signal processing technique associated with
deconvolution of measured signals after filtering through a known dynamic
system. Kitanidis and others extended this to the simultaneous estimation of
the input signal and the state of the intervening system. This is normally
posed as a special least-squares estimation problem with unbiasedness. The
approach has application in signal analysis and in control. Despite the
connection to optimal estimation, the standard algorithms are not necessarily
stable, leading to a number of recent papers which present sufficient
conditions for stability. In this paper we complete these stability results in
two ways in the time-invariant case: for the square case, where the number of
measurements equals the number of unknown inputs, we establish exactly the
location of the algorithm poles; for the non-square case, we show that the
best sufficient conditions are also necessary. We then draw on our previous
results interpreting these algorithms, when stable, as singular Kalman filters
to advocate a direct, guaranteed stable implementation via Kalman filtering.
This has the advantage of clarity and flexibility in addition to stability. En
route, we decipher the existing algorithms in terms of system inversion and
successive singular filtering. The stability results are extended to the time-
varying case directly to recover the earlier sufficient conditions for
stability via the Riccati difference equation.

###### keywords:

stability, state estimation, input estimation, singular filtering

##  1 Introduction

Simultaneous Input and State Estimation (SISE) algorithms take a system with
an unknown disturbance input sequence, $\\{d_{t}\\}$, and measured output
signal, $\\{y_{t}\\}$, plus possibly also a known input sequence,
$\\{u_{t}\\}$, and produce estimates of both $d_{t}$ and the system state
$x_{t}$, based on fixed lag smoothing. There have been a great number of
recent papers on these algorithms, with a recent subset [1, 2, 3, 4] focusing
on conditions for stability. A feature of SISE algorithms is that the
disturbance signals have uncertain provenance. So SISE presumes that no model
is available for this signal and the algorithm proceeds without an explicit
description of the disturbance signals’ statistical properties. Thus
techniques such as extended state observers [5] and augmented Kalman filters
[6, 7] are inapplicable, as both rely on disturbance models. A recent paper by
the authors [8] establishes that, when stable, the linear system SISE
algorithm of [9] coincides with the Kalman filter with $\\{d_{t}\\}$ modeled
by white Gaussian noise of unbounded variance. Various approaches consider
first estimating the state [1] and then using the state recursion to
reconstruct $d_{t}$, or estimating the disturbance first and then
reconstructing the state [9, 10, 4]. These methods rely on geometric
approaches and system inversion, although there is a strong overlap with
least-squares state estimation concepts of unbiasedness and optimality.

SISE algorithms go back to least to Kitanidis [11] with antecedents [12, 13,
14] concentrating on input signal reconstruction. Here, we follow the
formulation from Yong, Zhu and Frazzoli [4], which in turn builds on [9, 10].
We consider linear time-invariant systems to add clarity and to explore the
connection to optimal estimation before extending to uniformly time-varying
systems.

Input estimation is a signal processing technique associated with
deconvolution of measured signals after filtering through a known dynamic
system. Examples include the estimation of rainfall given river flow and the
calculation of salinity in the ocean accommodating for sensor dynamics [15].
Here, the central objective is to estimate the driving disturbance signal
$d_{t}$ and there is little interest in the sensor state. The algorithm should
be stable, however. Our particular driving problem, on the other hand, is the
estimation of generator states in part of a power grid when the
interconnection signals are unknown [16]. Here the priority is to estimate
network generator states in the face of unmodeled and unmeasured consumption,
which is treated as the disturbance signal. In spite of these distinct
objectives, the same algorithms have been used.

### Contributions & organization

Our objective in this paper is to attempt to bring some clarity and unity to
this picture by establishing precisely the connection to system inversion and
optimal estimation by deriving necessary and sufficient conditions for
stability using explicit system inverse formulæ and algebraic Riccati
equations, starting with the time-invariant case. Earlier stability conditions
were sufficient only but derived in the time-varying situation. We recover
these. Further, when stability is not achieved by these SISE algorithms, we
propose a modification based on inner-outer factorization, which maintains
state estimation performance at the expense of simple disturbance recovery.
This can be compared with the techniques advanced in [1] for approximate
system inversion with delay. Beyond this work, we know of no other which
addresses estimation when the stability conditions fail.

Section 2 presents the SISE problem for a linear time-invariant system.
Section 3 studies the zero direct feedthrough case and the corresponding SISE
of [9] and shows that, in the square case where the number of measurements
equals the number of disturbance channels, the input estimator is the inverse
of the $d_{t}$-to-$y_{t}$ system and the state estimator is a plant
simulation. Stability depends on the transmission zeros of the former system.
These necessary and sufficient stability conditions then are extended to the
non-square case with more measurements. This involves the Riccati difference
equation and a detectability condition. Section 4 expands this analysis to the
full-rank direct feedthrough case and comments on the non-full-rank case of
[4]. Section 5 draws connections to earlier works of singular filtering and
introduces an accommodation to circumvent stability issues using the inner-
outer factorization. It also contains the extension to time-varying systems
via the Riccati equation. Section 6 reinforces the connections to system
inversion and concludes. The Appendix contains the proofs.

##  2 Problem statement

SISE algorithms have been formulated for linear time-varying systems [9, 10,
4] and for nonlinear time-varying systems [15, 17]. However for clarity in
development, we consider the linear, time-invariant system with zero known
control input,

$\displaystyle x_{t+1}$ $\displaystyle=Ax_{t}+Gd_{t}+w_{t},$ (1)
$\displaystyle y_{t}$ $\displaystyle=Cx_{t}+Hd_{t}+v_{t},$ (2)

with $x_{t}\in\mathbb{R}^{n},$ $d_{t}\in\mathbb{R}^{m},$
$y_{t}\in\mathbb{R}^{p}.$ Zero-mean white noises $\\{w_{t}\\}$ and
$\\{v_{t}\\}$ are independent and independent from $\\{d_{t}\\}$ and $x_{0}$.
The covariance of $w_{t}$ is $Q\geq 0$ and the covariance of $v_{t}$ is $R>0.$
Denote the signal measurements
$\mathbf{Y}^{t}\triangleq\\{y_{t},y_{t-1},\dots,y_{0}\\}.$ The aim is to
produce from $\mathbf{Y}^{t},$ a recursive filtered state estimate,
$\hat{x}_{t|t},$ and filtered and/or smoothed estimates, $\hat{d}_{t|t+1}$ or
$\hat{d}_{t|t}$, depending on the properties of $G$ and
$\begin{bmatrix}C&H\end{bmatrix}$. We make the following assumption.

######  Assumption 1.

System (1-2) has $[A,C]$ observable, rank $G=m$, $[A,Q]$ reachable, and $R>0$.

Full-rank direct feedthrough, i.e. rank$(H)=m$, is treated in [10]; zero
direct feedthrough, $H=0$, in [9] with rank$(CG)=m$; and, [4] provides a
generalization, ULISE, with mixed rank properties between $H$ and $CG$. A
noise-free variant is treated in [1].

##  3 Zero direct feedthrough

For $H=0$ in (2), SISE from [9] is the recursion.

$\displaystyle X_{t}$ $\displaystyle=AP_{t-1}A^{T}+Q,$ (3) $\displaystyle
K_{t}$ $\displaystyle=X_{t}C^{T}(CX_{t}C^{T}+R)^{-1},$ (4) $\displaystyle
M_{t}$ $\displaystyle=[G^{T}C^{T}(CX_{t}C^{T}+R)^{-1}CG]^{-1}$
$\displaystyle\hskip 42.67912pt\times G^{T}C^{T}(CX_{t}C^{T}+R)^{-1},$ (5)
$\displaystyle P_{t}$ $\displaystyle=(I-K_{t}C)\left[(I-GM_{t}C)X_{t}\right.$
$\displaystyle\hskip
22.76219pt\left.\times(I-GM_{t}C)^{T}+GM_{t}RM_{t}^{T}G^{T}\right]$
$\displaystyle\hskip 51.21495pt+K_{t}RM_{t}^{T}G^{T},$ (6)
$\displaystyle\hat{d}_{t-1|t}$
$\displaystyle=M_{t}(y_{t}-CA\hat{x}_{t-1|t-1}),$ (7)
$\displaystyle\hat{x}_{t|t}$
$\displaystyle=A\hat{x}_{t-1|t-1}+G\hat{d}_{t-1|t}+K_{t}$ $\displaystyle\hskip
22.76219pt\times(y_{t}-CA\hat{x}_{t-1|t-1}-CG\hat{d}_{t-1|t}).$ (8)
$\displaystyle\mbox{\rm cov}(x_{t}|$ $\displaystyle\mathbf{Y}^{t})=P_{t},$ (9)

under the following structural condition.

######  Assumption 2.

$\displaystyle\text{rank}\,CG=m.$ (10)

An immediate observation is that SISE contains no specific information related
to a model for the unmeasured disturbance $d_{t}.$ Indeed, it is frequently
claimed that signal $\\{d_{t}:t=0,1,\dots\\}$ possesses no model whatsoever.
Although, for bounded covariance $X_{t}$, i.e. when the algorithm is stable,
the authors derived this version of SISE in [8] as a Kalman filter with
$\\{d_{t}\\}$ modeled as a white noise process of unbounded variance. We shall
return to this point later. Evidently, Assumption 2 requires $p\geq m$ and
rank $C\geq$ rank $G=m$. Firstly, we treat the square case, $p=m$, where the
number of measurements equals the dimension of the disturbance input. Then we
shall derive more general results.

###  3.1 Square zero-feedthrough case

From Assumption 2 when $p=m$, $CG$ is invertible. Since, from (5), $M_{t}CG=I$
or $M_{t}=(CG)^{-1}$, we have

$\displaystyle\hat{d}_{t-1|t}$
$\displaystyle=(CG)^{-1}(y_{t}-CA\hat{x}_{t-1|t-1}),$ (11) $\displaystyle 0$
$\displaystyle=y_{t}-CA\hat{x}_{t-1|t-1}-CG\hat{d}_{t-1|t},$ (12)
$\displaystyle\hat{x}_{t|t}$
$\displaystyle=A\hat{x}_{t-1|t-1}+G\hat{d}_{t-1|t},$ (13)
$\displaystyle=[I-G(CG)^{-1}C]A\hat{x}_{t-1|t-1}+G(CG)^{-1}y_{t}.$ (14)

This estimation algorithm:

  * –

is time-invariant;

  * –

does not depend on $Q$ or $R$, the noise variances;

  * –

is independent from the covariance calculations.,

  * –

has zero $\hat{x}_{t|t}$ innovations (12), (8).

SISE reduces to (11-14).

$\displaystyle\hat{x}_{t|t}$
$\displaystyle=[I-G(CG)^{-1}C]A\hat{x}_{t-1|t-1}+G(CG)^{-1}y_{t},$
$\displaystyle\hat{d}_{t-1|t}$
$\displaystyle=-(CG)^{-1}CA\hat{x}_{t-1|t-1}+(CG)^{-1}y_{t}.$

Note that, using the matrix inversion lemma, we may rewrite the SISE
$y_{t}$-to-$\hat{d}_{t-1|t}$ transfer function as

$\displaystyle(CG)^{-1}$ $\displaystyle\hskip
11.38109pt-(CG)^{-1}CA(zI-A+G(CG)^{-1}CA)^{-1}G(CG)^{-1}$
$\displaystyle=\left[CG+CA(zI-A)^{-1}G\right]^{-1},$
$\displaystyle=\left[zC(zI-A)^{-1}G\right]^{-1}.$ (15)

The filtered state estimate error satisfies

$\displaystyle\tilde{x}_{t|t}$ $\displaystyle\triangleq x_{t}-\hat{x}_{t|t},$
$\displaystyle=[I-G(CG)^{-1}C]A\tilde{x}_{t-1|t-1}$ $\displaystyle\hskip
14.22636pt+[I-G(CG)^{-1}C]w_{t-1}-G(CG)^{-1}v_{t}.$

The stability of SISE, i.e. the boundedness of the covariance of
$\tilde{x}_{t|t},$ depends on the eigenvalues of $[I-G(CG)^{-1}C]A$.

######  Theorem 1.

For system (1-2) with $p=m$ and subject to Assumption 2, the eigenvalues of
the SISE estimator system matrix, $\left[I-G(CG)^{-1}C\right]A,$ lie at the
transmission zeros of the square transfer function $zC(zI-A)^{-1}G.$
Accordingly, the SISE estimator is asymptotically stable if and only if these
transmission zeros all lie inside the unit circle.

The proof of this theorem follows immediately from (15). An alternate is given
in the Appendix for completeness and to establish connections to singular
filtering. We note that condition (10) in Assumption 2 implies that
$zC(zI-A)^{-1}G$ possesses exactly $n$ finite transmission zeros with exactly
$m$ at zero.

We see that, in the square case, the poles of SISE can be located precisely at
the transmission zeros of the $d_{t}$-to-$y_{t}$ transfer function. SISE
therefore is performing system inversion to recover $\hat{d}_{t-1|t}$ from
$\mathbf{Y}^{t}$. The dependent recursion (13) for $\hat{x}_{t|t}$ is a
simulation of the state equation (1) driven by $\hat{d}_{t-1|t}$. Effectively
all the information in $\mathbf{Y}^{t}$ is used in generating the disturbance
estimate, leaving simulation (13) to generate the state estimate.

When SISE is stable, it was shown in [8] that the state estimation algorithm
implements a Kalman filter with a model for $\\{d_{t}\\}$ as a white noise of
unbounded variance, $D$. In this case, the state estimation problem has
driving noise variance $Q+GDG^{T}$ and measurement noise variance $R$. The
identical filter, but not the covariances, will be achieved by taking driving
noise $GDG^{T}$ for finite $D$ and $R\to 0$. That is, SISE is a singular
filter. The connection to [18] in the proof is to the equivalent result in
Loop Transfer Recovery for LQG control. When one selects $R=0$, as opposed to
$R\to 0$ from above, then the poles are placed at the transmission zeros. The
limiting operation, on the other hand places the poles at the stable
transmission zeros and the inverses of the unstable transmission zeros [19].

###  3.2 Non-square zero-feedthrough case

From Assumption 2, we take $p\geq m$ and make a transformation of the output
signal as follows. This is a variation on the technique of [4]. Take the
singular value decomposition of $p\times m$ $CG$.

$\displaystyle\texttt{svd}(CG)$ $\displaystyle=U\Sigma V^{T},$
$\displaystyle=\begin{bmatrix}U_{m}&U_{p-m}\end{bmatrix}\begin{bmatrix}\Sigma\\\
0\end{bmatrix}V^{T}.$

Define the $p\times p$ transformation

$\displaystyle\mathcal{T}$
$\displaystyle=\begin{bmatrix}U_{m}^{T}-U_{m}^{T}RU_{p-m}(U_{p-m}^{T}RU_{p-m})^{-1}U_{p-m}^{T}\\\
U_{p-m}^{T}\end{bmatrix},$ (16)

and transform the original output signal, call it $\bar{y}_{t}$,

$\displaystyle y_{t}$
$\displaystyle=\mathcal{T}\bar{y}_{t}=\begin{bmatrix}C_{1}\\\
C_{2}\end{bmatrix}x_{t}+\begin{bmatrix}v_{1,t}\\\ v_{2,t}\end{bmatrix},$ (17)

yielding

$\displaystyle\det\,C_{1}G\neq
0,\;\;\;\;C_{2}G=0,\;\;\;\;\text{cov}\begin{bmatrix}v_{1,t}\\\
v_{2,t}\end{bmatrix}=\begin{bmatrix}R_{1}&0\\\ 0&R_{2}\end{bmatrix}.$

######  Theorem 2.

For system (1-2) with $p\geq m$ and subject to Assumptions 1 and 2, if and
only if the pair $[A(I-G(C_{1}G)^{-1}C_{1}),C_{2}]$ is detectable then the
filtered state covariance, $P_{t}$, is bounded and converges to a limit
$P_{\infty}$ as $t\to\infty.$

The corresponding gain matrices, $K_{\infty}$ and $M_{\infty}$, yield the
limiting SISE system matrix, $(I-K_{\infty}C)(I-GM_{\infty}C)A,$ with all its
eigenvalues strictly inside the unit circle.

The proof of this result appears in the Appendix and is based on proving that
the state covariance satisfies a Riccati Difference Equation. Although this
condition is not strictly the same as the condition in [4], the theorem
condition implies theirs. Hence, their condition is also necessary. Theorem 2
similarly extends the condition in [2]. The sufficient stability result in
[17] is predicated on $P_{t}$ being bounded a priori. We already know from
Theorem 1 the eigenvalues of $A(I-G(C_{1}G)^{-1}C_{1})$ are stable if and only
if $zC(zI-A)^{-1}G$ is minimum-phase.

We see that, when $p>m$, the surfeit of measurements beyond those strictly
needed to produce $\hat{d}_{t-1|t}$ are brought to bear on estimating $x_{t}$.
The stability of SISE depends on either the square case yielding stability via
Theorem 1, i.e. via stable transmission zeros, or there being sufficient
information in the additional measurements to stabilize the estimator.

When SISE is stable, then the algorithm implements a singular filter, as
explained above. However, now the corresponding singular filter is partially
singular, a term introduced in [20]. That is, the process noise variance is
finite but the measurement noise variance, $R,$ is less than full rank rather
than zero. The approach of [20], under the banner of stable optimal filtering,
in this case involves precisely a succession of a singular estimator and
followed by a regular estimator, as in SISE. The result in [8] derives this
stable (partially) singular filter when the plant satisfies the conditions of
Theorem 2.

##  4 Nonzero direct feedthrough

When $H\neq 0$ in (2), SISE alters. Gillijns and De Moor [10] provide a SISE
algorithm, subject to the following, for the time-invariant case.

######  Assumption 3.

Rank $H=m$.

Subject to this assumption, the SISE formulation for time-invariant system
(1-2) is

$\displaystyle\hat{x}_{t|t-1}$
$\displaystyle=A\hat{x}_{t-1|t-1}+G\hat{d}_{t-1|t-1},$ (18) $\displaystyle
P^{x}_{t|t}$
$\displaystyle=\begin{bmatrix}A&G\end{bmatrix}\begin{bmatrix}P^{x}_{t-1|t-1}&P^{xd}_{t-1|t-1}\\\
P^{dx}_{t-1|t-1}&P^{d}_{t-1|t-1}\end{bmatrix}\begin{bmatrix}A^{T}\\\
G^{T}\end{bmatrix}+Q,$ $\displaystyle\tilde{R}_{t}$
$\displaystyle=CP^{x}_{t|t-1}C^{T}+R,$ $\displaystyle M_{t}$
$\displaystyle=(H^{T}\tilde{R}_{t}H)^{-1}H^{T}\tilde{R}_{t}^{-1},$
$\displaystyle\hat{d}_{t|t}$ $\displaystyle=M_{t}(y_{t}-C\hat{x}_{t|t-1}),$
(19) $\displaystyle P^{d}_{t|t}$ $\displaystyle=(H^{T}\tilde{R}_{t}H)^{-1},$
$\displaystyle K_{t}$ $\displaystyle=P^{x}_{k|k-1}C^{T}\tilde{R}_{t}^{-1},$
$\displaystyle\hat{x}_{t|t}$
$\displaystyle=\hat{x}_{t|t-1}+K_{t}(y_{t}-C\hat{x}_{t|t-1}-H\hat{d}_{t|t}),$
(20) $\displaystyle P^{x}_{t|t}$
$\displaystyle=P^{x}_{t|t-1}-K_{t}(\tilde{R}_{t}-HP^{d}H^{T})K_{t}^{T},$
$\displaystyle P^{xd}_{t|t}$
$\displaystyle=\left(P_{t|t}^{dx}\right)^{T}=-K_{t}HP^{d}_{t|t}.$

When rank $H<m,$ [4] provide ULISE, a carefully developed SISE algorithm which
uses the singular value decomposition as in Subsection 3.2 but more widely to
handle the more complicated interaction between filtered and smoothed
estimates for $d_{t}$.

###  4.1 Square full-rank case

As with the $H=0$ case, we consider first rank $H=m$ and $m=p.$ That is $H$ is
invertible and, since $M_{t}H=I$, $M_{t}=H^{-1}$. Then SISE reduces to the
recursion

$\displaystyle\hat{d}_{t|t}$ $\displaystyle=H^{-1}(y_{t}-C\hat{x}_{t|t-1})$
$\displaystyle 0$ $\displaystyle=y_{t}-C\hat{x}_{t|t-1}-H\hat{d}_{t|t},$ (21)
$\displaystyle\hat{x}_{t+1|t}$
$\displaystyle=A\hat{x}_{t-1|t-1}+G\hat{d}_{t-1|t-1},$ (22)
$\displaystyle=(A-GH^{-1}C)\hat{x}_{t|t-1}+GH^{-1}y_{t},$
$\displaystyle\tilde{x}_{t+1|t}$
$\displaystyle=(A-GH^{-1}C)\tilde{x}_{t|t-1}+w_{t}-GH^{-1}v_{t}.$

######  Theorem 3.

For system (1-2) subject to Assumption 3, the eigenvalues of the SISE
estimator system matrix, $A-GH^{-1}C,$ lie at the transmission zeros of the
square transfer function $H+C(zI-A)^{-1}G.$ Accordingly, the SISE estimator is
asymptotically stable if and only if these transmission zeros all lie inside
the unit circle.

Proof:Applying the matrix inversion lemma to the square transfer function
between $d_{t}$ and $y_{t}$,

$\displaystyle\left[H+C(zI-A)^{-1}G\right]^{-1}$ $\displaystyle\hskip
28.45274pt=H^{-1}-H^{-1}C(zI-A+GH^{-1}C)^{-1}GH^{-1}.$

The poles of the square direct feedthrough SISE lie at the transmission zeros
of the $d_{t}$ to $y_{t}$ transfer function. $\Box$

Again, this result adds necessity to that of [4] in this case. Further, the
result does not rely on optimality arguments. As in the square zero
feedthrough case, the SISE estimator is time-invariant and independent from
$Q$ and $R$, and the state estimate filter innovations is zero. The condition
rank $H=m$ ensures that all $n$ transmission zeros are finite. We note again
that the state innovations sequence (21) is zero and the filter (22) simulates
$\hat{x}_{t+1|t}$ from $\hat{d}_{t|t}.$

###  4.2 Non-square full-rank case

The careful derivation of ULISE to accommodate rank $H\leq m$ is a central
contribution of [4] and involves separation into subspaces. Take the singular
value decomposition of $p\times m$ $H$ possessing rank $r$.

$\displaystyle\texttt{svd}(H)$ $\displaystyle=U\Sigma V^{T},$
$\displaystyle=\begin{bmatrix}U_{r}&U_{p-r}\end{bmatrix}\begin{bmatrix}\bar{H}&0\\\
0&0\end{bmatrix}V^{T}.$

Matrices take on the $(r,p-r)$ structure.

$\displaystyle H$ $\displaystyle=\begin{bmatrix}\bar{H}&0\\\
0&0\end{bmatrix},\;\;\;C=\begin{bmatrix}C_{1}\\\
C_{2}\end{bmatrix},\;\;G=\begin{bmatrix}G_{1}&G_{2}\end{bmatrix},$
$\displaystyle K_{t}$
$\displaystyle=\begin{bmatrix}K_{1,t}&K_{2,t}\end{bmatrix},\;\;\;M_{t}=\begin{bmatrix}M_{1,t}&M_{2,t}\end{bmatrix}.$

As earlier in (16) and (17), define the $p\times p$ transformation

$\displaystyle\mathcal{T}$
$\displaystyle=\begin{bmatrix}U_{r}^{T}-U_{r}^{T}RU_{p-r}(U_{p-r}^{T}RU_{p-r})^{-1}U_{p-r}^{T}\\\
U_{p-r}^{T}\end{bmatrix},$

and transform the original output signal, call it $\bar{y}_{t}$,

$\displaystyle y_{t}$
$\displaystyle=\mathcal{T}\bar{y}_{t}=\begin{bmatrix}\bar{C}_{1}\\\
\bar{C}_{2}\end{bmatrix}x_{t}+\begin{bmatrix}\bar{H}\\\
0\end{bmatrix}d_{t}+\begin{bmatrix}\bar{v}_{1,t}\\\
\bar{v}_{2,t}\end{bmatrix}.$

yielding $\det\,\bar{H}\neq 0$ and $\text{cov}\begin{bmatrix}\bar{v}_{1,t}\\\
\bar{v}_{2,t}\end{bmatrix}=\begin{bmatrix}\bar{R}_{1}&0\\\
0&\bar{R}_{2}\end{bmatrix}.$ When rank $H=m$, $\bar{H}$ is $m\times m$ and we
have the following result stemming from $M_{1,t}\bar{H}=I_{m}$.

######  Theorem 4.

Subject to Assumptions 1 and 3, $p\geq m,$ SISE with feedthrough is stable if
only if $[A-G\bar{H}^{-1}\bar{C}_{1},\bar{C}_{2}]$ is detectable.

The proof of this result is in the Appendix. This extends the detectability
condition111Note that [4] uses $p$ to denote our $m$. of Theorem 5 of [4] to a
necessary and sufficient condition for stability of SISE in this case. It also
is the analog of Theorem 2 for the full-rank feedthrough case.

An alternative way to view necessity is to write the system matrix of SISE as

$\displaystyle A-[AK_{t}+(G-AK_{t}H)M_{t}]C$
$\displaystyle=A-G\bar{H}^{-1}\bar{C}_{1}$ $\displaystyle\hskip
28.45274pt-(AK_{2,t}+GM_{2,,t}-AK_{1,t}\bar{H}M_{2,t})\bar{C}_{2}.$

For this matrix to be stable, a multiple of $\bar{C}_{2}$ must stabilize
$A-G\bar{H}^{-1}\bar{C}_{1}.$

###  4.3 Less than full rank feedthrough

We build again on the decomposition above of [4] and make the following
assumption.

######  Assumption 4.

$\text{rank}\,\bar{C}_{2}G_{2}=m-\text{rank}\,\bar{H}.$

In [4], the authors derive a sufficient condition for stability which we now
extend to necessity.

######  Theorem 5.

Subject to Assumptions 1 and 4, general feedthrough SISE is stable if only if
$[A-G_{1}\bar{H}^{-1}\bar{C}_{1},\bar{C}_{2}]$ is detectable.

This detectability condition is shown in [4] to be sufficient for stability by
using the filter recursion for $\hat{x}_{t|t}$. If one calculates the
alternative recursive prediction, $\hat{x}_{t|t-1},$ then it is evident that
the ULISE system matrix is again of the form

$\displaystyle(A-G_{1}\bar{H}^{-1}\bar{C}_{1})(I-\tilde{L}_{t}\bar{C}_{2})(I-G_{2}M_{2,t}\bar{C}_{2})$
$\displaystyle=A-G_{1}\bar{H}^{-1}\bar{C}_{1}+W_{t}\bar{C}_{2},$

for appropriate $W_{t}$. Evidently, this can be stable only if the
detectability condition holds.

##  5 Upshots

###  5.1 Stability and singular filtering

The preceding analysis provides necessary and sufficient conditions for the
stability of linear SISE algorithms. Further, for the square cases, it yields
the precise locations on the algorithm poles and demonstrates that the
emphasis is on $d_{t}$-to-$y_{t}$ system inversion to recover $d_{t}$ followed
by best efforts to estimate the state. We have pointed out the successive
estimation nature of SISE, as have others. The question remains as to actions
to be taken when SISE proves to be unstable, noting these central properties:

  1. (i)

SISE is stable when the $d_{t}$-to-$y_{t}$ system is stably invertible.

  2. (ii)

When SISE is stable, it corresponds (at least in the zero feedthrough case) to
a singular Kalman filter.

  3. (iii)

Subject to: detectability of $[A,C]$, stabilizability of
$[A,Q^{\frac{1}{2}}]$, and $R\stackrel{{\scriptstyle+}}{{\to}}0$; the singular
Kalman filter is a stable estimator by construction.

  4. (iv)

The stability conditions for the singular Kalman filter are more relaxed than
Assumptions 2, 3 or 4.

  5. (v)

When SISE proves to be unstable, it differs from the Kalman filter.

###  5.2 Kalman filtering for input and state estimation

Denote the transfer function from $d_{t}$ to $y_{t}$ by $T(z)$. As derived in
earlier sections, when $zT(z)$ has all its transmission zeros inside the unit
circle, then SISE is guaranteed stable and is equivalent to a specific stable
singular Kalman filter. Compute a discrete-time inner-outer
factorization222Strictly speaking, this is a co-inner-co-outer factorization
because of the ordering of $T_{i}$ and $T_{o}$ [21]. It can be obtained from
the inner-outer factorization of $T^{T}$. [22, 23].

$\displaystyle T(z)$ $\displaystyle=T_{o}(z)T_{i}(z),$

with $m\times m$ $T_{i}(z)$ an inner function, i.e. stable and all-pass, and
$p\times m$ $T_{o}(z)$ an outer function, i.e. all transmission zeros inside
the unit circle.

We note the following from the construction of the inner-outer factors.

######  Lemma 1.

If $T(z)$ has realization

$\displaystyle x_{t+1}$ $\displaystyle=Ax_{t}+Bu_{t}+Gd_{t}+w_{t},$
$\displaystyle y_{t}$ $\displaystyle=Cx_{t}+Du_{t}+Hd_{t}+v_{t},$

then $T_{o}(z)$ has realization

$\displaystyle x_{t+1}$
$\displaystyle=Ax_{t}+Bu_{t}+\check{G}\check{d}_{t}+w_{t},$ $\displaystyle
y_{t}$ $\displaystyle=Cx_{t}+Du_{t}+\check{H}\check{d}_{t}+v_{t},$

where $\check{d}_{t}$ is the output of $T_{i}(z).$

That is, for the same initial conditions, $T_{o}(z)$ and $T(z)$ have the same
states and have realizations which differ only in the $G$ and $H$ matrices.
This factorization is depicted in Figure 1.

Figure 1: System inner-outer factorization

Applying SISE or the singular Kalman filter for the outer function $T_{o}(z)$
to the signal $\\{y_{t}\\}$ yields estimates of $\\{\check{d}_{t}\\}$ and
$\\{x_{t}\\}$ via a stable algorithm with guaranteed statistical and
optimality properties. This follows since $T_{o}$ is stably invertible by
construction. Further, this stability depends on the standard assumptions
above for Kalman filter stability. If one uses SISE, then depending on the
delay properties of $T_{o}(z)$, i.e. its behavior as $z\to\infty$, a modified
variation of the algorithm and Assumption 2, 3, or 4 might be needed to
accommodate $d_{t}$-to-$y_{t}$ invertibility. This is discussed further in
Section 6.

To recover estimates for original system inputs $\\{d_{t}\\}$ from those for
$\\{\check{d}_{t}\\}$ requires deconvolution (input estimation) without state
estimation for the maximum-phase but stable system $T_{i}$. If delay is not an
issue, then this can proceed stably via a fixed-interval smoother or reverse-
time input estimation.

If the state estimates of $x_{t}$ themselves are the objective, then the
reconstruction of $\check{d}_{t}$ versus $d_{t}$ is immaterial. This is the
nature of the problem addressed in partially-known power system state
estimation [16].

The singular filters derived by Shaked and co-authors [19, 20, 24] rely on the
Return Difference Equality and spectral factorization for their calculation.
In the case where the transmission zeros are unstable, the filter solution
replaces them by their inverses akin to the inner-outer factorization.

The Kalman filter of [8] for $T_{o}(z)$ may be derived from the state-space
model below with appropriate covariances,

$\displaystyle\begin{bmatrix}x_{t+1}\\\ \check{d}_{t+1}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}A&\check{G}\\\
0&0\end{bmatrix}\begin{bmatrix}x_{t}\\\
\check{d}_{t}\end{bmatrix}+\begin{bmatrix}I_{n}&0\\\
0&I_{m}\end{bmatrix}\begin{bmatrix}w_{t}\\\ \delta_{t}\end{bmatrix},$
$\displaystyle y_{t}$
$\displaystyle=\begin{bmatrix}C&\check{H}\end{bmatrix}\begin{bmatrix}x_{t}\\\
\check{d}_{t}\end{bmatrix}+v_{t},$

or using the direct construction as in [8], which avoids an explicit model for
$d_{t}$ but yields the same filter.

Marro and Zattoni [1] provide guidance on the recovery of the disturbance
input signal when the $d_{t}$-to-$y_{t}$ system is non-minimum-phase. Their
approach involves the approximate inversion of this system using a long delay
to accommodate the nominal instability of this inverse. Such techniques are
reminiscent of those advanced in [25]. While the approach in [1] centers on
state-estimation first, their development is geometric and noise free and so,
it is unclear how this affects performance. Of course, the geometric analysis
throws up the same initial reliance on minimum-phase zeros for stability and
exact inversion.

###  5.3 Extension to time-varying systems

Developments so far have been limited to the time-invariant case and have
availed themselves of concepts of transmission zeros, stable invertibility and
inner-outer factorization, each of which is problematic to extend to time-
varying systems. However, since alternative results have been phrased for the
time-varying case, we consider this extension now, relying on examination of
SISE recursions via Riccati difference equations in the proofs of Theorems 2
and 4.

Appealing to [9, 10] for the time-varying SISE algorithms in the case of
Theorem 2 and zero direct feedthrough, Riccati equation (27) becomes

$\displaystyle X_{t+1}$
$\displaystyle=\bar{A}_{t}X_{t}\bar{A}_{t}^{T}-\bar{A}_{t}X_{t}C_{2,t}^{T}(C_{2,t}X_{t}C_{2,t}^{T}+R_{2,t})^{-1}$
$\displaystyle\hskip
96.73936pt\times(\bar{A}_{t}X_{t}C_{2,t}^{T})^{T}+\bar{Q}_{t},$

where,

$\displaystyle\bar{A}_{t}$
$\displaystyle=A_{t}(I-G_{t-1}(C_{1,t}G_{t-1})^{-1}C_{1,t}),$
$\displaystyle\bar{Q}_{t}$
$\displaystyle=A_{t}G_{t-1}(C_{1,t}G_{t-1})^{-1}R_{1,t}(G_{t-1}(C_{1,t}G_{t-1})^{-1})^{T}A_{t}^{T}$
$\displaystyle\hskip 22.76219pt+Q_{t},$

and, in the case of full-rank feedthrough, (7) becomes

$\displaystyle X_{t+1}$
$\displaystyle=\hat{A}_{t}X_{t}\hat{A}_{t}^{T}-(\hat{A}_{t}X_{t}\bar{C}_{2,t}^{T})(\bar{C}_{2,t}X_{t}\bar{C}_{2,t}+\bar{R}_{2,t})^{-1}$
$\displaystyle\hskip
96.73936pt\times(\hat{A}_{t}X_{t}\bar{C}_{2,t}^{T})^{T}+\hat{Q}_{t},$

where,

$\displaystyle\hat{A}_{t}=A_{t}-G_{t}\bar{H}_{t}^{-1}\bar{C}_{1,t},\;\;\;\hat{Q}_{t}=Q_{t}+G_{t}\bar{H}_{t}^{-1}\bar{R}_{1,t}\bar{H}_{t}^{-T}G_{t}^{T}.$

with now time-varying quantities $\\{A_{t},G_{t},\dots,\\}$. We may appeal to
standard sufficient results, e.g. [26, 6] and Theorem 5.3 in [27], on the
exponential stability of the Kalman filter subject to uniform reachability and
detectability. Subject to the uniform satisfaction of time-varying equivalents
of Assumptions 1, 2 and/or 3 as appropriate, this extends these stability
conditions to the uniformly time-varying case.

##  6 System Inversion and SISE

For the square cases of SISE satisfying Assumptions 1 or 2, we were able to
demonstrate that the SISE $d_{t}$-estimator implements exactly the left
inverse of the $y_{t}$-to-$d_{t}$ system. The simultaneous $x_{t}$-estimate is
the state of the inverse system the stability of which depends on the
transmission zeros of the original system.

Conditions for left invertibility of a linear time-invariant system are
provided by Sain and Massey [28] and for stable invertibility by Moylan [29]
via the Rosenbrock system matrix. Both papers construct the inverse system.
Moreover, in [28, 30], left invertibility with delay, $L$, is studied, where
stacked measurements
$\begin{bmatrix}y_{t}^{T}&y_{t+1}^{T}&\dots&y_{t+L}^{T}\end{bmatrix}^{T}$ are
used to estimate $d_{t}$. Marro and Zattoni blend into this picture stable
approximate inversion with delay.

From [28], we see that, for any $p\geq m$:

  * –

Assumption 2 is the left invertibility condition for $C(zI-A)^{-1}G$ with
delay one.

  * –

Assumption 3 is the left invertibility condition for $H+C(zI-A)^{-1}G$ with
delay zero.

  * –

Assumption 4 is the left invertibility condition for $H+C(zI-A)^{-1}G$ with
delay one.

Given the input recovery objective of SISE, this is not surprising. But it is
interesting to tie these ideas more closely.

It is worth remarking that many presentations of SISE algorithms make
connections to ‘unbiasedness’ and ‘optimality’ of the state estimate. As [1,
8] demonstrate, the probabilistic concept of unbiasedness is really tied to a
geometric property of the algorithms and the nature of certain subspaces. The
optimality of the state estimates is within the class of estimators already
satisfying the geometric constraints. As is evident from, say, Theorems 1 and
2 and the Riccati equation proof, there is no degree of freedom left for the
state estimator in the square case and limited degrees of freedom in the non-
square case. Indeed, in the square cases, (13-22) show that $\hat{x}_{t|t}$ is
computed by system simulation using the estimated input; the measurements play
no further part. The detectability conditions on Theorems 2 and 4 show how the
remaining degrees of freedom are used in the Riccati difference equations (28)
and (7).

In conclusion, the paper attempts to unify the collection of SISE algorithms
by revealing their explicit connections to system inversion to recover the
otherwise unmodeled disturbance input $d_{t}$ followed by their ‘best efforts’
subsequent estimation of the state $x_{t}$. The result has been to develop
necessary and sufficient conditions for stability, at least in the linear
time-invariant case, in terms of the transmission zeros of the
$d_{t}$-to-$y_{t}$ plant and then the detectability of the subsequent state
estimator.

##  7 Appendix

### Proof of Theorem 1

We loosely follow a calculation from Maciejowski [18]. From (14) and the
system equations the (time-invariant) transfer function from $d_{t}$ to
$\hat{x}_{t|t}$ via $y_{t}$ is given by

$\displaystyle\Psi(z)$
$\displaystyle=\left\\{zI-\left[I-G(CG)^{-1}C\right]A\right\\}^{-1}zG(CG)^{-1}$
$\displaystyle\hskip 85.35826pt\times C(zI-A)^{-1}G,$ (23)
$\displaystyle=\left\\{zI-\left[I-\Pi\right]A\right\\}^{-1}z\Pi(zI-A)^{-1}G,$

where we have used $\Pi\triangleq G(CG)^{-1}C$. Write

$\displaystyle\left[zI-(I-\Pi)A\right]^{-1}z\Pi$
$\displaystyle=\left[zI-(I-\Pi)A\right]^{-1}\left[z\Pi-zI+(I-\Pi)A\right]+I,$
$\displaystyle=-\left[zI-(I-\Pi)A\right]^{-1}(I-\Pi)(zI-A)+I.$

Then, since $(I-\Pi)G=0,$

$\displaystyle\Psi(z)=$
$\displaystyle-\left[zI-(I-\Pi)A\right]^{-1}(I-\Pi)(zI-A)(zI-A)^{-1}G$
$\displaystyle\hskip 85.35826pt+(zI-A)^{-1}G,$
$\displaystyle=-\left[zI-(I-\Pi)A\right]^{-1}(I-\Pi)G+(zI-A)^{-1}G,$
$\displaystyle=(zI-A)^{-1}G.$ (24)

From (23), $\Psi(z)$ is the product of two transfer functions and nominally
should have $2n$ poles; those at the eigenvalues of $A$ and those at the
eigenvalues of $(I-\Pi)A$. The transfer function $zC(zI-A)^{-1}G$ has McMillan
degree $n$ with $n$ finite transmission zeros. We see from (24) that only
poles at the eigenvalues of $A$ are present in $\Psi$. This implies that the
poles due to the eigenvalues of $(I-\Pi)A$ cancel the transmission zeros of
$zC(zI-A)^{-1}G.$

### Proof of Theorem 2

Define the following quantities.

$\displaystyle\mathcal{H}_{t}$
$\displaystyle=GM_{t}-K_{t}CGM_{t}+K_{t},\;\;\;Z=C_{1}G,$ $\displaystyle Y$
$\displaystyle=(CX_{t}C^{T}+R)^{-1}=\begin{bmatrix}Y_{1}&Y_{2}\\\
Y_{2}^{T}&Y_{3}\end{bmatrix},$

where $Y$ is divided conformably with $C$ and $v_{t}$ in (17).

From (7) and (8) the filtered prediction error satisfies

$\displaystyle\tilde{x}_{t}$ $\displaystyle\triangleq x_{t}-x_{t|t}$
$\displaystyle=(I-\mathcal{H}_{t}C)A\tilde{x}_{t-1}+(I-\mathcal{H}_{t}C)w_{t-1}-\mathcal{H}_{t}v_{t}.$

Whence,

$\displaystyle P_{t|t}$
$\displaystyle=\operatorname{cov}(x_{t}|\mathbf{Y_{t}})$
$\displaystyle=(I-\mathcal{H}_{t}C)(AP_{t-1|t-1}A^{T}+Q)(I-\mathcal{H}_{t}C)^{T}$
$\displaystyle\hskip 22.76219pt+\mathcal{H}_{t}R\mathcal{H}_{t}^{T}.$

Using (3) yields

$\displaystyle X_{t+1}$
$\displaystyle=A\left((I-\mathcal{H}_{t}C)X_{t}(I-\mathcal{H}_{t}C)^{T}+\mathcal{H}_{t}R\mathcal{H}_{t}^{T}\right)A^{T}$
$\displaystyle\hskip 142.26378pt+Q.$ (25)

We show that this discrete Lyapunov equation is also a Riccati difference
equation by substituting for $\mathcal{H}_{t}$ using
$CG=\begin{bmatrix}Z^{T}&0\end{bmatrix}^{T}$.

$\displaystyle\mathcal{H}_{t}$
$\displaystyle=G\left(\begin{bmatrix}Z^{T}&0\end{bmatrix}Y\begin{bmatrix}Z\\\
0\end{bmatrix}\right)^{-1}\begin{bmatrix}Z^{T}&0\end{bmatrix}Y-X_{t}\begin{bmatrix}C_{1}^{T}&C_{2}^{T}\end{bmatrix}$
$\displaystyle\hskip 22.76219ptY\begin{bmatrix}Z\\\
0\end{bmatrix}\left(\begin{bmatrix}Z^{T}&0\end{bmatrix}Y\begin{bmatrix}Z\\\
0\end{bmatrix}\right)^{-1}\begin{bmatrix}Z^{T}&0\end{bmatrix}Y$
$\displaystyle\hskip
34.1433pt+X_{t}\begin{bmatrix}C_{1}^{T}&C_{2}^{T}\end{bmatrix}Y$
$\displaystyle=\begin{bmatrix}GZ^{-1}&GZ^{-1}Y_{1}^{-1}Y_{2}\end{bmatrix}+X_{t}\begin{bmatrix}0&C_{2}(Y_{3}-Y_{2}^{T}Y_{1}^{-1}Y_{2})\end{bmatrix},$
(26)

Using partitioned matrix inversion with $Y$ gives

$\displaystyle(Y_{3}-Y_{2}^{T}Y_{1}^{-1}Y_{2})$
$\displaystyle=(C_{2}X_{t}C_{2}^{T}+R_{2})^{-1}$ $\displaystyle
Y_{1}^{-1}Y_{2}$
$\displaystyle=-(C_{1}X_{t}C_{2}^{T})(C_{2}X_{t}C_{2}^{T}+R_{2})^{-1}.$

Substituting this into (26) and (25) gives the following Riccati difference
equation.

$\displaystyle X_{t+1}$
$\displaystyle=\bar{A}X_{t}\bar{A}^{T}-(\bar{A}X_{t}C_{2}^{T})(C_{2}X_{t}C_{2}^{T}+R_{2})^{-1}$
$\displaystyle\hskip 96.73936pt\times(\bar{A}X_{t}C_{2}^{T})^{T}+\bar{Q},$
(27)

where,

$\displaystyle\bar{A}$ $\displaystyle=A(I-G(C_{1}G)^{-1}C_{1}),$
$\displaystyle\bar{Q}$
$\displaystyle=AG(C_{1}G)^{-1}R_{1}(G(C_{1}G)^{-1})^{T}A^{T}+Q.$ (28)

Appealing to Theorem 14.3.1 [31] (p. 510), provided
$[\bar{A},\bar{Q}^{\frac{1}{2}}]$ is stabilizable and $[\bar{A},C_{2}]$ is
detectable, then $X_{t}$ converges to the maximal solution of the algebraic
Riccati equation, which is stabilizing.

Now, since by assumption $[A,Q^{\frac{1}{2}}]$ is stabilizable, there exists a
$\mathcal{K}$ such that $A-Q^{\frac{1}{2}}\mathcal{K}$ is stable. Taking,

$\displaystyle\bar{Q}^{\frac{1}{2}}$
$\displaystyle=\begin{bmatrix}Q^{\frac{1}{2}}&AG(C_{1}G)^{-1}R^{\frac{1}{2}}\end{bmatrix},$

and
$\bar{\mathcal{K}}=\begin{bmatrix}\mathcal{K}^{T}&R^{\frac{T}{2}}\end{bmatrix}^{T}$,
$\bar{A}-\bar{Q}^{\frac{1}{2}}\bar{\mathcal{K}}$ also is stable. So
stabilizability of $[A,Q^{\frac{1}{2}}]$ implies stabilizability of
$[\bar{A},\bar{Q}^{\frac{1}{2}}]$.

### Proof of Theorem 4

The proof parallels that of Theorem 2. Substitute (19) and (20) into (18) to
yield

$\displaystyle\hat{x}_{t+1|t}$
$\displaystyle=(A-\mathcal{L}_{t}C)\hat{x}_{t|t-1}+\mathcal{L}_{t}y_{t},$

where $\mathcal{L}_{t}=AK_{t}-AK_{t}HM_{t}+GM_{t}$. Then

$\displaystyle\tilde{x}_{t+1|t}$ $\displaystyle\triangleq
x_{t}-\hat{x}_{t+1|t},$
$\displaystyle=(A-\mathcal{L}_{t}C)\tilde{x}_{t|t-1}+w_{t}-\mathcal{L}_{t}v_{t},$
$\displaystyle X_{t+1}$
$\displaystyle=(A-\mathcal{L}_{t}C)X_{t}(A-\mathcal{L}_{t}C)^{T}+\mathcal{L}_{t}R\mathcal{L}_{t}^{T}+Q,$

with $X_{t+1}\triangleq\operatorname{cov}(x_{t+1}|\mathbf{Y}^{t})$. Dividing
$K_{t}$ and $M_{t}$ conformably with $C^{T}$:
$K_{t}=\begin{bmatrix}K_{1,t}&K_{2,t}\end{bmatrix},$
$M_{t}=\begin{bmatrix}M_{1,t}&M_{2,t}\end{bmatrix},$ one arrives directly at
the following Riccati difference equation.

$\displaystyle X_{t+1}$
$\displaystyle=\hat{A}X_{t}\hat{A}^{T}-\hat{A}X_{t}\bar{C}_{2}^{T}(\bar{C}_{2}X_{t}\bar{C}_{2}+\bar{R}_{2})^{-1}\bar{C}_{2}X_{t}\hat{A}^{T}$
$\displaystyle\hskip 113.81102pt+\hat{Q},$ (29)

where,

$\displaystyle\hat{A}=A-G\bar{H}^{-1}\bar{C}_{1},\;\;\;\hat{Q}=Q+G\bar{H}^{-1}\bar{R}_{1}\bar{H}^{-T}G^{T}.$

The proof follows as that for Theorem 2 using [31].

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# A $d^{8}$ anti-Hund’s Singlet Insulator in an Infinite-layer Nickelate

Hyo-Sun Jin1 Warren E. Pickett2<EMAIL_ADDRESS>Kwan-Woo Lee1,3
<EMAIL_ADDRESS>1Division of Display and Semiconductor Physics, Korea
University, Sejong 30019, Korea
2Department of Physics and Astronomy, University of California Davis, Davis CA
95616, USA
3Department of Applied Physics, Graduate School, Korea University, Sejong
30019, Korea

###### Abstract

The status of nickelate superconductors in relation to cuprate high
temperature superconductors is one of the concepts being discussed in high
temperature superconductivity in correlated transition metal oxides. New
additions to the class of infinite layer nickelates can provide essential
input relating to connections or distinctions. A recently synthesized compound
Ba2NiO2(AgSe)2, which contains isolated ‘infinite layer’ NiO2 planes, may lead
to new insights. Our investigations have discovered that, at density
functional theory mean field level, the ground state consists of an unusual
$e_{g}$ singlet on the Ni2+ ion arising from large but separate Mott
insulating gaps in both $e_{g}$ orbitals, but with different, anti-Hund’s,
spin directions of their moments. This textured singlet incorporates at the
least new physics, and potentially a new platform for nickelate
superconductivity, which might be of an unconventional form for transition
metal oxides due to the unconventional undoped state. We include in this paper
a comparison of electronic structure parameters of Ba2NiO2(AgSe)2 with a
better characterized infinite layer nickelate LaNiO2. We provide more analysis
of the $d^{8}$ anti-Hund’s singlet that emerges in this compound, and consider
a minimally correlated wavefunction for this singlet in an itinerant
background, and begin discussion of excitations – real or virtual – that may
figure into new electronic phases.

##  I Introduction

The discovery of superconductivity, after 30 years of attempts, in hole-doped
‘infinite layer’ nickelates ${\cal R}$NiO2 (${\cal R}$=lanthanide)H.Hwang2019
, has broadened the scope of high temperature superconductivity (HTS). Such
nickelates have identical crystal structures and underlying $d^{9}$ metal atom
configurations, yet nickelate superconductivity has been remarkably elusive.
Although the achieved temperatures so farH.Hwang2020 ; Ariando2020 , Tc up to
15 K, are not yet HTS, the connection is clear. The nickelates have the same
square planar transition metal-oxygen layers, and the same underlying formal
oxidation state with a single hole in the $3d$ shell: Ni1+ rather than Cu2+.
In both systems, doping the $d^{9}$ configuration toward $d^{8}$ leads to
superconductivity.H.Hwang2020 ; Ariando2020  Yet with these and other
similarities, essential differences have been uncovered lee2004 ; choi_prr2020
; arita2021 ; wen2021 ; review_our2021 .

An early study by some of the authors of the ‘infinite layer’ compound LaNiO2
(Ni1+, $d^{9}$) revealed substantial differences which emphasized that, while
relations between the two ions are evident, Ni1+ does not behave like Cu2+ in
these structures lee2004 . When strong correlation effects are included in the
band calculation, there was a strong tendency toward the formation of peculiar
on-site ‘singlet’ spin character in a metallic background arising from spin
compensation between the two $e_{g}$ orbitals. This behavior arises in spite
of the formal spin-half (not spin $S$=0 or $S$=1) criterion of the formal
oxidation state. The carriers at the Fermi level (and below – the hole states)
consisted of roughly equal $d_{x^{2}-y^{2}}$ spin up, $d_{z^{2}}$ spin down.
This incipient singlet leaves oxygen ions straying from the O2- configuration,
forming the metallic background. In the same sort of calculation for Cu2+, a
spin-half antiferromagnetic (AFM) insulating state is obtained, consistent
with experiment.

Another distinction from cuprates is that a $d^{8}$ nickelate Ba2NiO2(AgSe)2
(BNOAS) has been reported bnoas_exp , and it contains an infinite-layer NiO2
layer. The formally $d^{8}$ cuprate LaCuO3 is a conventional although
metastable metal, with samples being oxygen deficient karpinen  and typically
prepared under pressure. Other formally $d^{8}$ cuprates are rare. The
magnetic character of BNOAS is enigmatic. Conventionally, the $d^{8}$
configuration should be a non-magnetic singlet, with two holes in the
$d_{x^{2}-y^{2}}$ orbital, or a triplet, with spin parallel holes in each of
the $e_{g}$ orbitals. The former has zero (spin) susceptibility, and the
latter would have a Curie-Weiss susceptibility with AFM ordering at some
temperature, supposing conventional antialigned neighbors.

What was observed by Matsumoto et al. bnoas_exp  is (when zero field cooled) a
constant value, on top of which is a rounded peak centered at T∗=130 K with
half width $\sim$10 K – apparently some magnetic reconstruction in an
otherwise magnetically inert state. When field cooled, the susceptibility is
higher below T∗, and a weak peak is seen at ($\frac{1}{2},\frac{1}{2}$,0) in
polarized neutron scattering at 5 K, suggesting antialigned neighboring
moments. Matsumoto et al. suggested a canted AFM state of $S$=1 spins to
account for the field cooling dependence below T∗, but other aspects of the
susceptibility remain to be explained. These are some of the questions we
address.

The report of BNOAS has reinvigorated study of the two dimensional (2D) Kondo
necklace model bnoas_ours , the 1D version of which was proposed by Doniach
Doniach . The 2D model consists of a square lattice of two-orbital sites, with
in-plane coupling $J$ between neighboring orbital1 states and on-site Kondo
coupling $K$ between orbital1 and orbital2. Wu et al. have applied the iPEPS
(infinite projected entangled pair state) formalism to obtain the phase
diagram of the anisotropic XXZ version tu_bnoas , finding spin-singlet,
supersolid, and Bose-Einstein condensed phases for various ratios $K/J$.
Finite temperature studies by Singh RRPSingh  included the local $S=1$ excited
states that are exchange coupled to neighboring $S=1$ states. This spin model
leads to a ground state first-order transition between singlet and
antiferromagnetic phases, with finite temperature behavior being amenable to
cluster expansion techniques. Separately, Kitamine et al. have pursued design
of $d^{8}$ nickelate materials kitamine2020 , focusing on
$A_{2}$NiO${}_{2}X_{2}$ compounds with $A$ being a divalent alkaline earth
cation and anions $X$=H, F, Cl, Br, I, the aim being to isolate the
$d_{x^{2}-y^{2}}$ orbital and leave other $d$ orbitals just below the Fermi
energy rather than to engineer magnetic behavior.

The organization of the paper is as follows. Section II provides the input to
the theoretical work: crystal structure and computational methods. The non-
magnetic electronic structure, including a comparison of $d^{8}$ and $d^{9}$
ions in infinite layer environments, is provided in Sec. III, and the magnetic
calculations are introduced. In Sec. IV effects of correlations are
introduced, and the peculiar band singlet that arises is discussed and its
origin analyzed. Then some analysis of this anti-Hund’s singlet and a
corresponding correlated Hartree-Fock wavefunction is given in Secs. V and VI,
with the intent that it might shed light on the peculiar observed magnetic
behavior of BNOAS. A Discussion and Summary is provided in Sec. VII.

Figure 1: (a) Body-centered tetragonal structure of Ba2NiO2(AgSe)2. (b)
Tetragonal Brillouin zone doubled along the $\hat{c}$-axis and high symmetry
points given in the band structure plots below.

##  II Crystal Structure and Methods

BNOAS crystal structure. BNOAS is usefully considered as [Ba(AgSe)2Ba]NiO2 –
an active $\infty$-layer Ni2+O2 plane separated by thick Ba(AgSe)2Ba blocking
layers bnoas_exp . The space group is body-centered tetragonal $I4/mmm$ (#139)
bnoas_exp , and the Ni ion has $4/mmm$ symmetry. As displayed in Fig. 1(a),
the (AgSe)2 blocking layer has the FeSe structure: Ag in the central layer is
coordinated tetrahedrally by Se atoms, which in the next layers are neighbored
by a plane of Ba2+ ions. Ni has no apical oxygen neighbors.

The compound is apparently metastable: it was synthesized under 7 GPa pressure
at 850 ∘C. This is reminiscent of the superconducting ‘infinite layer’
nickelates, which are only superconducting when grown in strained perovskite
form using layer-by-layer synthesis techniques H.Hwang2019 , with O later
removed topotactically. In BNOAS, strain supplied by the (AgSe)2 layer leads
to an anomalously large Ni-O separation of 2.105 Å, which Matsumoto et
al.bnoas_exp  argued should put BNOAS beyond an $S=0$ to $S=1$ transition at
2.05 Å. Our earlier results also indicated a change in electronic behavior
bnoas_ours  around 2.05 Å.

Theoretical methods. Our calculations are based at the density functional
theory (DFT) level using the generalized gradient approximation (GGA) gga
implemented in wien2k wien2k . In wien2k, we used the same input parameters in
our previous studies of BNOAS bnoas_ours . The in-plane lattice constant is
$a$=3.96 (4.21) Å for LaNiO2 (BNOAS). The latter anomalously large value is
believed to be key to the unusual behavior of the Ni ion bnoas_exp ;
bnoas_ours .

Correlation effects were included as conventionally done using the DFT+U
approach ylvisaker2010  as in Ref. bnoas_ours , using GGA for the semi-local
exchange-correlation effects. In DFT+U, Hund’s coupling was fixed at $J=0.7$
eV. For the magnetic calculations on BNOAS, the on-site Coulomb repulsion
$U$=$7$ eV was chosen for most of the studies. We did check that results are
fairly insensitive to somewhat smaller values of $U$. For example, at $U=5$
eV, the hopping parameters differed by at most 10 meV. The energy gap (which
is not the Mott-Hubbard gap, see following sections) is small in this range of
$U$, closing around $U_{c}\approx$5.5 eV (specifically, $U_{c}$= 5.2 eV and
5.8 eV in fplo (Full Potential Local Orbital)fplo  and wien2k, respectively).
We note that in wien2k $U$ and $J$ are applied within an atomic sphere. The
sphere does not include quite all of the atomic orbital (whose tails are not
well defined anyway), also the atomic function is distinct from the Wannier
function (see below) that might be argued to be a more realistic ‘local
orbital’. Both of these aspects would require, and perhaps justify, a somewhat
smaller value of ‘atomic’ $U$ to achieve the same impact of on-site repulsion.

To construct the maximally localized Wannier functions, we used the wannier90
wan90  and wien2wannier wien2wan  codes, projecting onto symmetry adapted
Wannier functions. Two frozen energy windows were investigated: –5.7 eV to 1.3
eV (–7 eV to 3.5 eV) for 27 (37) starting orbitals, given below, for
nonmagnetic (respectively magnetic, textured singlet) BNOAS. The parameters
for BNOAS, included in Table I, are obtained for Ni $d$, O $p$, Se $(s),p$,
and Ag $(s,p),d$ orbitals (the orbitals inside parentheses are the additional
orbitals used for the singlet state).

The excellent quality of the Wannier bands for NM BNOAS are shown in Fig. 2;
the corresponding result for magnetic BNOAS is displayed in Appendix A. For
comparison, we also calculated Wannier parameters for NM LaNiO2. For a fit
with more flexibility compared with the previous results of Botana and Norman
botana2020  where Ni $d$, O $p$, and La only $d_{z^{2}}$ orbitals were used,
we used 23 starting orbitals of La $d,f$, Ni $d$, and O $p$ orbitals within an
energy widow of –8.5 eV to 1.8 eV. The excellent representation of the bands
is shown in Appendix A.

Figure 2: A full valence band Wannier fit to the band structure for NM BNOAS;
the two are indistinguishable. As an initial starting point, the following 27
orbitals were used: $d$ orbitals of Ni and Ag ions, and $p$ orbitals of O and
Se ions.

##  III Electronic and Magnetic Structure

Table 1: On-site energies and hopping parameters (in units of eV), obtained from the transformation to Wannier functions using 27 (37) orbitals in the unit cell (see text), for NM ($U=0$) and singlet ($U=7$ eV) states of BNOAS. NM LaNiO2 is provided for comparison. The O1 [$\frac{1}{2}$,0,0] and O2 [0,$\frac{1}{2}$,0] ions sit along the $\hat{a}$\- and $\hat{b}$-direction, respectively. Only near-neighbor hoppings are listed, others for further neighbors are much smaller although necessary for excellent fits. Boldface and italics denote values for specific attention. Here, $\Delta_{cf}$, $\delta_{cf}$, and $\delta^{p}_{cf}$ indicate the crystal field splittings between the centers of $t_{2g}$ and $e_{g}$ orbitals, between $e_{g}$ orbitals, and between $p_{\sigma}$ and $p_{\pi}$ orbitals, respectively.  On-site energies |  LaNiO2 | BNOAS
---|---|---
|  NM |  NM | Singlet
|  |  |  up |  dn
$d_{xy}$ | -1.62 | -1.10 | -3.84 | -4.36
$d_{xz,yz}$ | -1.46 | -0.79 | -4.57 | -4.24
$d_{x^{2}-y^{2}}$ | -1.04 | -0.74 | -5.27 | 2.62
$d_{z^{2}}$ | -1.45 | -0.67 | 2.43 | -5.61
$\Delta_{cf}$ | 0.27 | 0.19 |  |
$\delta_{cf}$ | 0.41 | -0.07 |  |
$p_{x}=p_{\sigma}$ O1 | -4.85 | -3.27 | -2.56 | -2.38
$p_{y}=p_{\pi}$ O1 | -3.34 | -2.06 | -1.97 | -2.01
$p_{z}$ O1 | -3.39 | -2.09 | -1.49 | -1.49
$p_{x}=p_{\pi}$ O2 | -3.34 | -2.06 | -1.97 | -2.01
$p_{y}=p_{\sigma}$ O2 | -4.85 | -3.27 | -2.56 | -2.38
$p_{z}$ O2 | -3.39 | -2.09 | -1.49 | -1.49
$\delta^{p}_{cf}$ | 1.51 | 1.21 |  |
Hopping |  |  |  |
amplitudes |  |  |  |
$d_{xy}-p_{\pi}$ O1 | -0.65 | -0.53 | -0.48 | -0.47
$d_{xz}-p_{z}$ O1 | -0.70 | 0.56 | 0.48 | 0.48
$d_{x^{2}-y^{2}}-p_{\sigma}$ O1 | 1.25 | 0.97 | 0.98 | 1.03
$d_{z^{2}}-p_{\sigma}$ O1 | -0.46 | -0.28 | -0.70 | -0.68
$p_{y}$ O2 $-p_{x}$ O1 | -0.44 | -0.51 | -0.08 | -0.09
$p_{x}$ O2 $-p_{x}$ O1 | 0.16 | -0.22 | -0.19 | -0.20
$p_{y}$ O2 $-p_{y}$ O1 | 0.16 | -0.22 | -0.19 | -0.20
$p_{x}$ O2 $-p_{y}$ O1 | -0.23 | -0.11 | -0.13 | -0.13
$p_{z}$ O2 $-p_{z}$ O1 | -0.15 | -0.01 | 0.07 | 0.09

###  III.1 $d^{8}$ versus $d^{9}$ configuration

Many of the aspects of the electronic and magnetic structures of BNOAS were
presented in our earlier paper bnoas_ours , and referring to Fig. 4 of that
paper will facilitate the reading of this section. A different perspective is
gained from identifying how the BNOAS and ${\cal R}$NiO2 electronic structures
of the NiO2 electronic parameters differ in the Ni $d$ \- O $p$ energy range,
given the same infinite layer scheme (no apical oxygen) with somewhat
differing environments. The on-site energies and hopping amplitudes obtained
from Wannier function representations are presented in Table 1 for ${\cal
R}$=La. Also shown are the differences between NM and the magnetic BNOAS
parameters, to indicate the changes that result from applying $U$=7 eV for
correlation effects.

Orbital site energies: nonmagnetic case. On the broadest level, one can
consider the mean position of all five $d$ levels relative to the Fermi level
$E_{F}$, something that is not often considered because the $t_{2g}$ states
are so strongly bound and inert. The mean $3d$ site energies are -0.82 in
BNOAS, -1.41 in LaNiO2, indicating an overall “chemical shift” upward of 0.6
eV in BNOAS of the Ni site. This shift will be due largely to the difference
of nearby cation, Ba2+ in BNOAS versus La3+ in LaNiO2, with some difference
arising the different formal charge states.

The $e_{g}$ crystal subsplitting,
$\delta_{cf}=\varepsilon_{x^{2}-y^{2}}-\varepsilon_{z^{2}}$, is reduced
drastically and changed in sign in BNOAS (-0.07 eV) compared to 0.41 eV in
LaNiO2. This LaNiO2 character is common in cuprates, leading to many
considerations as a single-band system at zero-th order. It is highly unusual
that $d_{z^{2}}$ should lie at or above the $d_{x^{2}-y^{2}}$ energy, although
the difference in site energies in BNOAS is nearly negligible. These
differences will assume importance in later analysis, especially the effective
degeneracy of the two $e_{g}$ orbitals. However, bandwidths (next section)
need also to be considered.

The charge transfer (CT) energy $\varepsilon_{d}-\varepsilon_{p}$ receives
close attention in transition metal oxides. From the mean Ni $d$ and O $p$
site energies, the CT energies are 2.45 eV for LaNiO2, 1.65 eV for BNOAS. This
difference leaves BNOAS nearer the CT insulator – Mott-Hubbard (MH) insulator
crossover, in between the MH insulator nickelates and the CT insulator
cuprates. LaNiO2 itself is usually considered as closer or in fact in the MH
insulator regime berciu2020 . Of course, the ${\cal R}$NiO2 infinite layer
nickelates have pockets of carriers of other character lee2004 ; choi2020 ,
thus are considered by many as self-doped Mott insulators ZYZ2020 ;
lechermann2020 ; karp2020 .

Hopping amplitudes. The $d$-$p$ hopping amplitudes are of special interest
since, with the site energies, they determine band positions and widths and
are of interest in evaluating exchange couplings. Since the $p$ bands are
fully occupied, the differences in $p$-$p$ hopping are not of interest here,
but are included in Table 1 for reference. As in cuprates, the in-plane
hopping $t_{dp\sigma}$ is largest ($\sim 1$ eV), but not necessarily dominant.
Some $d$-$p$ hoppings are quite different between the two compounds. The
hopping $d_{x^{2}-y^{2}}$-$p_{\sigma}$ = $t_{dp\pi}$ differs by 25%: 1.25
(LaNiO2) versus 0.97 (BNOAS). Other $d$-$p$ hoppings differ by less. These
differences affect the shape and the bandwidths of the important bands, which
have $e_{g}$ character. One interesting feature is that magnetism increases
the $d_{z^{2}}-p_{\sigma}$ hopping amplitude by a factor of 2.5.

We note that some site energies and hopping amplitudes are sensitive to what
orbitals are included in the Wannier basis set and the energy window. This
sensitivity should be taken into account when comparing these quantities
between different practitioners.

###  III.2 DFT band structure

The tight-binding parameters discussed in the previous section can give an
incomplete picture of the electronic structure. The NM bands of BNOAS are
shown in Fig. 2. Whereas the two $e_{g}$ on-site energies are essentially the
same ($d_{z^{2}}$ is slightly higher), only the $d_{x^{2}-y^{2}}$ band extends
above EF. It does so by 2 eV, due to the usual large hopping
$t_{dp\sigma}\approx$1 eV. The $d_{z^{2}}$ bandwidth arises from twice hopping
$d_{z^{2}}$-$p_{x}$-$d_{z^{2}}$ with small amplitude, which (with other
mixtures) leaves the $d_{z^{2}}$ orbital roughly doubly occupied and
$d_{x^{2}-y^{2}}$ roughly doubly empty. LaNiO2 is much like cuprates, with its
$d_{x^{2}-y^{2}}$ orbital about half-filled (with complications due to the La
$d$ orbital dispersion).

Figure 3: Ni $d$ orbital-projected densities of states (DOSs) in (top) FM and
(bottom) AFM at the DFT level. These states are $S=1$ states, i.e. magnetic
Ni. Minority DOSs are plotted downward.

In the FM state, the total moment in GGA is 1.29 $\mu_{B}$/f.u., consisting of
atomic (sphere) moments of 1.04 for Ni, 0.07 for O, and 0.03 for Se (in units
of $\mu_{B}$). This reduction from 2$\mu_{B}$ reflects itinerant character and
also indicates significant $p-d$ hybridization accounting for reduction from a
naive Hund’s value. The AFM state has a nearly identical Ni local moment.

Considering spin-ordering, only $S$=$1$ states (i.e. magnetic Ni) are obtained
at the DFT level. Figure 3 shows the Ni $d$ orbital-projected DOSs for the AFM
and FM alignments. In the minority channel, the $d_{x^{2}-y^{2}}$ band is
fully unfilled and the $d_{z^{2}}$ band is partially empty, with the other
bands fully filled. Note that the minority $d_{z^{2}}$ band is well localized
around $E_{F}$ (narrow peak) in both FM and AFM alignments. In the majority
channel, most $d$ bands are nearly filled, but some of the dispersive
$d_{x^{2}-y^{2}}$ band crosses above $E_{F}$.

Figure 4: Left panel: The spin up and spin down bands of the singlet band
structure at $U=7$ eV. Colors and their thickness of the fatbands indicate the
degree of orbital character as denoted by the inset. Right Panels: (a) Hole
spin density isosurface plot containing the bottom of the conduction bands in
the range of 0.2 eV to 1.7 eV. In addition to the characters on the NiO2
layers, Se $p_{x}+p_{y}$ and Ag $s$ characters are visible. (b) Spin density
isosurface plot, which is the integral over the valence bands of the
difference of up and down spin densities. The oxygen participation in the CuO4
unit is evident (see text). In the isosurface plots, the isovalue is 0.015
$e$/Å3.  Figure 5: Virtual crystal band structures and atom-projected DOSs of
AFM ($X^{2.5+}$)2NiO2(AgSe)2 at the DFT level, showing an energy gap of 0.3
eV. The Ni configuration is $d^{9}$, with a minority $d_{x^{2}-y^{2}}$ hole.

##  IV Toward a $d^{9}$ state

###  IV.1 DFT+$U$ electronic structure

The correlated DFT+U band structures lead, as anticipated, to magnetic Ni
ions, with the bands shown in Fig. 4 (left panel). In this calculation all Ni
atoms have the same atomic configuration – the unit cell is not doubled. At
first look, the spin-up versus spin-down differences seem minor. This state
occurs even when applying a small $U$ such as $U_{eff}$=$U$-$J$$\approx 0.3$
eV with $J=0.7$ eV in the DFT+U approach bnoas_ours .

Further analysis reveals qualitative differences: there is spin polarization,
but the number of occupied up (or down) bands is the same, so the net moment
vanishes (the choice of ‘up’ [majority $d_{x^{2}-y^{2}}$] and ‘down’ was
arbitrary). Yet the system has FM rather than AFM symmetry; the latter has
natural symmetry dictated vanishing net moment in a doubled cell, and
identical up and down bands. Correlation has produced in BNOAS a state with
the symmetry of a FM – distinct up and down bands but no broken space group
symmetry – but vanishing moment. This result is indicative of an anti-Hund’s
magnetic state. Note that inclusion of $U$ also has increased the moment to
near the ideal Hund’s high-spin value of 2$\mu_{B}$/Ni for local-moment AFM or
FM order.

This symmetry analysis is reminiscent of a so-called half metallic AFM (or
compensated half-metal) wep01 ; wep08  but is insulating. What has occurred is
that Mott insulating behavior has emerged in both $d_{x^{2}-y^{2}}$ and
$d_{z^{2}}$ orbitals, but with spin-reversed occupation (or equivalently,
unoccupied) orbitals. The spin-half configurations of each orbital are coupled
to a singlet. The issue of classical (band) singlet versus quantum (correlated
orbital) difference is discussed in Secs. IV and V.

Due to other active orbitals in the gap region, the band gap is not the MH
gap. The gap-edge states involve Ba $d$, Se $s,p$, and Ag $s$ orbitals, and
these bands become somewhat magnetically polarized, differently for the up and
down spin directions. Similar bands in ${\cal R}$NiO2 materials overlap EF and
figure strongly in theoretical models.botana2020 ; berciu2020 ; choi2020 ;
lechermann2020 ; karp2020

The next step is to consider orbital character of bands more closely, and to
notice that for the up bands, and unlike cuprates, the unoccupied $d$ band
unexpectedly has $d_{z^{2}}$ character, while the down spin hole is in the
$d_{x^{2}-y^{2}}$ orbital. This canceling of spin-half moments in different
orbitals was denoted previously bnoas_ours  as an “off-diagonal singlet”, off-
diagonal in orbital giving it an anisotropic spin density with vanishing net
moment, yet having exchange coupling to neighbors through the Ni
$d_{x^{2}-y^{2}}$ orbitals. Our original report bnoas_ours  provided more
description of the electronic structure. This DFT+$U$ singlet is, to our
knowledge, unique in the condensed matter physics literature, although such
correlated magnetic states have long been studied by quantum chemists qchem .

###  IV.2 Virtual crystal electron doping

To investigate a $d^{9}$ system like ${\cal R}$NiO2, one Ba ion is replaced by
a La ion. Using the virtual crystal approximation, both Ba ions are replaced
by a virtual ion $X^{2.5+}$ with nuclear charge Z=56.5, restoring the lattice
symmetry and adding one electron/f.u. which goes to the Ni ion. In this
calculation, we used the lattice parameters of BNOAS bnoas_ours , but the
internal parameters are optimized with wien2k.

At the DFT level, the AFM state is energetically favored over FM and NM states
by 116 and 155 (in units of meV/Ni), respectively. In both spin-ordered
states, the Ni local moment is 0.84 $\mu_{B}$, consistent with a strong
$S=\frac{1}{2}$ spin reduced somewhat by hybridization, as is common. These
energy differences lead, in a fixed moment nearest neighbor Heisenberg
picture, to a superexchange $J=116$ meV. Comparing with values for infinite-
layer nickelates $J$=64 meV from spin wave theory leonov2020  and spectroscopy
Lu2021 , and the range of values reported for cuprates $J$=125-158 meV
cuprate2017j , this value for La-substituted BNOAS lies in an intermediate
range. The similarities suggest that electron-doped BNOAS is a candidate for
magnetically coupled superconductivity, perhaps with higher Tc than known
nickelate superconductors.

The AFM band structure is shown in Fig. 5. Since a gap opens already for
$U$=0, we show the results at the GGA level. At this virtual crystal level,
doped holes go primarily into Se $s,p$ bands along the $M$-$A$ line. This
prospect is much different than discussions of cuprates and related transition
metal oxides, where the question is MH insulator versus CT insulator and where
the respective holes reside. In BNOAS the blocking layer is becoming active,
receiving both electron- and hole-doped carriers. Prospects for
superconductivity are a topic for the future, but it seems clear that doped
BNOAS will be distinct in behavior from doped cuprates and most transition
metal oxides, with some potential similarity to infinite-layer nickelates.

##  V Spin polarization;
the anti-Hund’s singlet

The character of doped holes and their relationship to superconductivity in
HTS cuprates have been a central concern. The CT, versus MH, characterization
of the energy gap leads to a large amount of Zhang-Rice (ZR) singlet character
zhangrice  in doped holes and the electron removal spectrum. The ZR singlet is
a coupling of spin-half Cu spin with an oppositely directed linear combination
of planar O $p_{\sigma}$ orbitals of $d_{x^{2}-y^{2}}$ symmetry. Holes in a
Mott insulator would have simple $d_{x^{2}-y^{2}}$ character, with small
(negligible) O character. The $d^{8}$ Ni ion in BNOAS presents a different
reference state, yet singlet versus triplet character becomes a central
feature, albeit on-site rather than involving oxygen.

Including $U$ in the electronic structure calculation to account for on-site
repulsion in a mean field picture, and thereby obtaining magnetic character,
causes important changes in the electronic structure and the derived band
parameters, as shown in Table 1 under the two rightmost BNOAS columns. First,
it should be expected that all Ni $d$ orbitals will be shifted: strongly
downward if occupied, somewhat upward if unoccupied. These differences will
appear most apparently in the on-site parameters, but a secondary but still
significant effect can be seen in Wannier hopping amplitudes.

BNOAS presents in the calculated ground state not the trivial singlet of a
completely unoccupied $d_{x^{2}-y^{2}}$ orbital, but a singlet with internal
character as described in the previous section: there is exchange coupling
within the NiO2 layer and weaker coupling between layers, leading to the Kondo
sieve model bnoas_ours  for the insulating spin system. The results that we
present for BNOAS are for this singlet phase, which persists in our
calculations for $U>4$ eV, using two different all-electron DFT codes
wien2kwien2k  and fplofplo  and with various starting spin configurations. In
BNOAS, for each spin direction all five majority orbitals are occupied while
one minority orbital is unoccupied. This peculiar singlet that we find to be
persistent differs in which $e_{g}$ spin-orbital is unoccupied. Because the
spin splitting between majority and minority is so large, the mean $d$ site
energy in the magnetic state becomes less relevant.

For the Ni $t_{2g}$ orbitals, the in-plane $d_{xy}$ orbital is more strongly
bound than the other two by 0.3-0.4 eV, an effect of environment and bonding-
type hybridization in-plane. The $e_{g}$ orbitals are of primary interest. The
mean $e_{g}$ on-site energies (majority, minority; see Table I) are nearly
degenerate. Individually, the exchange (up minus down) site energies are
around 8 eV, equal to $U$=7 eV plus some contribution from Hund’s $J$ and
hybridization effects. The mean $d$ sites energies are 1.0-1.3 eV lower for
the singlet case, because there are more occupied majority orbitals (moved
down) than minority orbitals (moved up). Regarding oxygen, only the
$p_{\sigma}$ orbitals show a significant spin dependence, with majority being
0.2 eV lower than minority.

For the hopping amplitudes of BNOAS, only one shows a significant difference
due to singlet formation relative to the NM state. This one is the
$d_{z^{2}}$-$p_{\sigma}$ hopping, normally considered to be insignificant in
square-planar geometry. While it is -0.28 eV in NM BNOAS, in the singlet case
it becomes -0.7 eV for both spins. The effect is to spread $d_{z^{2}}$
character into band positions (bonding and antibonding) where it would
otherwise be negligible.

Origin of the Singlet. The underlying feature of significance of the GGA band
parameters for anti-Hund’s singlet formation is the effective degeneracy of
the two $e_{g}$ orbital energies. One could consider a unitary transformation
among the $e_{g}$ orbitals but reducing the symmetry without any evident
usefulness does not appear attractive. The anti-Hund’s orientation for spin-
half moments costs $J_{H}/2$ = 0.35 eV in exchange energy, for the value of
$J_{H}\approx J$=0.70 eV we have used in our DFT+$U$ calculations (see Sec.
II). The value of pair spin $S$=0 or $S$=1 (i.e. the relative spin density
orientation) affects the band structure and hence the kinetic energy, the
charge and spin densities, and the correlation energy, with some contribution
from the purely Coulombic (charge density) terms. The energy sources favoring
the anti-Hund’s orientation are not clear. It is worth noting that in LaNiO2,
where the $e_{g}$ subsplitting is of the usual sign with a value of 0.7 eV,
even then in the large $U$ regime a similar singlet character (opposite
orientation of $e_{g}$ spins) became dominant lee2004 . Studies varying the
lattice constant, hence the interatomic and interionic separations, could be
helpful in understanding the mechanism of singlet formation.

##  VI A quantum singlet in an itinerant background

###  VI.1 A minimally correlated wavefunction

The Kohn-Sham (KS) treatment within DFT provides an effective single-particle
(mean field) picture of the system. The ‘singlet’ discussed so far is the
classical-spin version of one spin up, one spin down on each Ni ion, with no
quantum nature of the state. A generalization to a quantum singlet in an
unpolarized background (at least in the atomic limit) can be represented by a
wavefunction that is a generalization of the KS or Hartree-Fock (HF) one:

$\Psi(\\{r_{i,s}\\})={\cal
A}~{}~{}Q(r_{1,\uparrow},r_{2,\downarrow})D^{\uparrow}(\\{r_{j,\uparrow}\\})D^{\downarrow}(\\{r_{j,\downarrow}\\}),$
(1)

Here $Q$ is the singlet wavefunction for particles 1 and 2, and $j$ runs only
over the other particle indices. $D^{\uparrow}$ ($D^{\downarrow}$) is the
Slater determinant for the occupied particle states (except for singlet
orbitals) with spin up (down). ${\cal A}$ is the antisymmetrization operator
which finally provides a function $\Psi$ that is antisymmetric under exchange
of any two particles’ coordinates. This wavefunction is heuristic, as the
issue of projecting out particle coordinates 1 and 2 out of the determinants
is a remaining problem. (Also, in actual application it would contain
unscreened exchange and no correlation.)

The determinants contribute nothing to the net spin polarization unless the
orbitals are spin-dependent. The HF states, for local orbitals
$\phi_{1},\phi_{2}$ involved in a singlet ($e_{g}$ orbitals for BNOAS), are
$\phi_{m}(1,s)=\phi_{m}(1)|s\rangle$ in terms of the spin or $|s\rangle$
($s=\pm 1$ for up and down respectively) and the argument of $\phi_{m}$ is
simplified to $\phi_{m}(1)$ for particle 1, etc. and the orbitals are the same
for both spins (by antisymmetry). The spin-singlet wavefunction is

$\displaystyle Q(1,2)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}([\phi_{1}(1)\phi_{2}(2)+\phi_{1}(2)\phi_{2}(1)]$
(2)
$\displaystyle\times\frac{1}{\sqrt{2}}[|\uparrow\downarrow\rangle-|\downarrow\uparrow\rangle]$

in common notation.

###  VI.2 Features of the wavefunction

The charge (+) and spin (-) density operators are

${\hat{n}}_{\pm}(r)=\sum_{i}[\delta(r-r_{i,\uparrow})\pm\delta(r-r_{i,\downarrow})].$
(3)

Taking the matrix elements in $|Q\rangle$, one obtains that
$n_{+}(r)$=$\phi^{2}_{1}(r)+\phi^{2}_{2}(r)$ (which are the KS or HF values)
and, from spin inversion of $Q$ giving only a change in sign, the spin density
$n_{-}(r)$=$m(r)$ is identically zero. Thus singlet formation does not change
the total density, and the spin density vanishes (at this level of
description) even for anisotropic orbitals $\phi_{m}$: the up amplitude of
each orbital cancels its down amplitude. Notice however from Eq. (2) that the
singlet wavefunction itself is textured, i.e. anisotropic, allowing exchange
coupling between sites. If the quantum nature of the singlet is quenched (as
in DFT theory) the non-zero spin polarization is of an incomplete singlet
type, similar to that seen earlier lee2004 .

The correlated HF-singlet wavefunction of Eq. (2) brings new terms into the
hybrid Kohn-Sham HF equations. The kinetic energy for a given particle
includes new ‘spin exchange’ terms

$\nabla^{2}\Psi=(\nabla^{2}Q){\cal D}+Q(\nabla^{2}{\cal
D})+2\vec{\nabla}Q\cdot\vec{\nabla}{\cal D}$ (4)

where ${\cal D}$ denotes the product of the two Slater determinants. The
singlet couples the spin directions, bringing in new ‘exchange’ energies from
$Q$ in the kinetic energy. Some alteration in the usual Coulomb exchange
potential will also arise from the singlet spin-mixing. Further analysis
requires more specificity of viewpoint, due to the substantial differences in
intent and in form of the HF and the KS equations, or a possible hybrid HF-KS
treatment.

###  VI.3 Excitations above the singlet phase

BNOAS presents interesting complications. Considering the correlated HF
wavefunction posited above, the actual existence of the singlet ground state
presents questions. With the spin-coupled $e_{g}$ orbitals removed from the
band structure (the Slater determinants) the calculated band structure (with
compensating classical spins) becomes approximate. In addition, one of the
orbitals, say $\phi_{1}$ ($d_{x^{2}-y^{2}}$) is coupled to neighboring
$\phi_{1}$ orbitals in-plane, favoring antialignment, while the $\phi_{1}$ and
$\phi_{2}$ orbitals are coupled to a singlet. Neglecting further interactions
(discussed by us earlier bnoas_ours ) this situation is a realization of the
Kondo necklace Hamiltonian, in the regime where Kondo coupling is dominant.

For weaker Kondo coupling the $S=1$ state can compete with the $S$=$0$
singlet, with some results in this regime having been explored by Singh
RRPSingh . The triplet nature of the ion, with orientations $M_{S}$=+1,0,-1,
and anticipated local anisotropy due to the environment, provides a rich
palette for study. Considering the unusual susceptibility (reviewed in the
Introduction) a favored $M_{S}$=0 orientation may be of special interest to
consider for BNOAS. With a small (calculated) bandgap, band excitations are
available to participate in optical conductivity and, virtually, in many-body
processes that impact low energy phenomena, including superconductivity.

##  VII Discussion and Summary

Spin singlet physics in 2D systems has played a large role in cuprate HTS, and
similarities of nickelates to cuprates have encouraged extension of this
concept to superconducting nickelates. The dominant concept has been that of
ZR singlet zhangrice , building on the characterization of undoped cuprates as
CT insulators rather than explicit MH insulators. The latter would involve
only $d$ orbital reoccupation by doping or low energy excitations, while in CT
insulators doped holes will lie primarily on neighboring O $p$ orbitals. A
linear combination on four O sites will form a molecular orbital of
($d_{x^{2}-y^{2}}$) symmetry, and a hole in this molecular orbital will pair
with the central spin-half Cu2+ to form a spin singlet, the ZR singlet. While
most studies have assigned nickelates as having more Mott than CT character,
the ZR singlet concept is sprinkled through the recent nickelate literature.
The infinite layer nickelates are more often considered from a Hund-Hubbard
viewpoint chang2019 . We find that electronic structure parameters suggest
that BNOAS lies in the vicinity of a charge transfer to Mott-Hubbard (CT-MH)
crossover, making the basic ‘physics’ harder to pinpoint.

Zhang, Yang, and Zhang (ZYZ) have proposed ZYZ2020  a generalized singlet
construct. Noting that even undoped NdNiO2 has a small concentration of Nd
$5d$ carriers mixing somewhat with Ni $d$ states, they have proposed that ZR-
like combinations of these carriers form a local molecular orbital and form a
singlet. Each occupied Nd singlet removes a spin from the Ni square lattice,
and it is proposed that the few percent provided by Nd inhibits AFM ordering
of the Ni spins. These ZYZ singlets might also account for observed low-T
transport behavior. The model was built on the Mott insulator viewpoint before
the CT insulator picture of nickelates became more prevalent. Photoemission
spectroscopies are now indicating charge or spin ordering Tam_CDW_2021 ;
Krieger_CDW-SDW_2021 ; Rossi_CO_2021  in ${\cal R}$NiO2 (${\cal R}$ a
lanthanide atom), complicating any connection between such singlets and
inhibition of geometrical symmetry breaking.

Singlet phases arise in the 2D Shastry-Sutherland spin model in certain
parameter regimes, but they have a different character and origin than those
discussed above. On a square lattice of $S$=$\frac{1}{2}$ spins, a specific
regime of exchange parameters can result in the formation and ordering of
singlets from spins on neighboring sites or across the diagonal of the square
Haravifard2014 . Such exchange processes enrich the phase diagram.

In this paper it has been shown by comparison in what ways electronic
structure parameters differ between $d^{8}$ BNOAS and $d^{9}$ LaNiO2. Both
share the infinite-layer local environment of the Ni ion, but the further
environment (Ba2+ versus La3+ etc.) affects several parameters. Separately,
the strongly increased in-plane a lattice constant, with related perpendicular
changes, has a strong influence on the physics.

##  VIII Acknowledgments

We acknowledge communications with A. S. Botana, M. R. Norman, and V. Pardo,
and conversations with R. R. P. Singh, and feedback from Wei-Lin Tu and Hyun-
Yong Lee. H.S.J. and K.W.L. were supported by National Research Foundation of
Korea Grant No. NRF-2019R1A2C1009588. W.E.P. was supported by NSF Grant No.
DMR 1607139.

Figure 6: Wannier fitting for the textured singlet phase of BNOAS at $U=7$ eV.
Dashed red lines indicate the fit to the calculated bands (solid black lines).
As the initial starting point for this fitting used the following 37 orbitals:
Ni {$d$}, O {$p$}, Se {$s$, $p$}, and Ag {$s,p,d$} orbitals. For this fit the
Se $s$ and Ag $s$, $p$ bands were added to include lower conduction band
hybridization.  Figure 7: Wannier fit of the bands for NM LaNiO2, providing
the parameters given in Table 1. For this fit, 23 starting orbitals of La
$d,f$, Ni $d$, and O $p$ were used.

##  Appendix A Wannier fits to
the Textured Singlet Bands

Figure 6 provides comparison of the band structures of the textured singlet
state of BNOAS. The comparison is for the correlated DFT+$U$ bands, spin up
and spin down (black lines), and the Wannier representation that includes 37
orbitals in the primitive cell (dashed red lines that mostly overlie the black
lines). The Wannier representation is excellent through the valence bands and
in the $d$ bands in the conduction bands. The lower conduction bands are not
representative of Mott insulating, basically localized orbital, states because
they include large dispersion departing from $\Gamma$ toward $Z$. The
dispersion arises from mixing with Se and Ag $s$,$p$ states in the same energy
region, through Ba and Ni $d_{z^{2}}$ orbitals. In the spin up bands, the
conduction Mott band would be flat around 1.2 eV except for the mixing. For
reference and comparison with Fig. 2, the non-magnetic bands and Wannier fit
for LaNiO2 are provided in Fig. 7.

This character is important in considering doping. Upon adding electrons, they
will lie in the (AgSe)2 blocking layer, itinerant and without much coupling to
the Ni spin. This holds for both spin directions, although the Mott conduction
band for spin down lies around 1 eV higher than for the up spin, and is not
easily discernible in the figure.

##  Appendix B Band structure of non-magnetic $(X^{2.5+}$)2NiO2(AgSe)2

Figure 8: (a) Virtual crystal band structure and atom-projected DOSs, and (b)
Fermi surface (FS) of NM ($X^{2.5+}$)2 NiO2(AgSe)2 in the DFT level. The FS is
a single cylinder centered at the $M$ point.

Our discussion of electron doping in Sec. IV.B was based on a virtual crystal
treatment of the cations in the system Ba2+La3+NiO2(AgSe)2, leading to
($X^{2.5+}$)2 NiO2(AgSe)2. Figure 8(a) displays the non-magnetic band
structure and atom-projected DOS of this system. The DOS can be compared with
that of the (preferred energy) AFM DOS in Fig. 5, and these bands provide a
useful reference for how magnetism affects the electronic structure giving the
bands of Fig. 5. The gap lies at the middle of the $d_{x^{2}-y^{2}}$ band,
leading to the classic $d^{9}$ half-filled band system in the 2D square
lattice. The opening of the AFM gap leads to the narrow $d_{z^{2}}$ bands
lying at the bottom of the gap, which is a central difference compared to
$d^{9}$ cuprate systems, usually discussed in terms of the charge transfer
energy difference.

The corresponding Fermi surface, shown in Fig. 8(b), consists of a single 2D
cylinder centered at the M point. The circular shape indicates an incipient 2D
nesting instability for scattering $|\vec{Q}|$=2$k_{F}$ ($k_{F}$ is the Fermi
wavevector) for the half-filled band, contributing to the gap that arises for
AFM ordering but not for FM order.

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