January, 2023

On higher-spin ${\cal N}=2$ supercurrent multiplets

Sergei M. Kuzenko and Emmanouil S. N. Raptakis

Department of Physics M013, The University of Western Australia
35 Stirling Highway, Perth W.A. 6009, Australia

Email<EMAIL_ADDRESS><EMAIL_ADDRESS>

We elaborate on the structure of higher-spin $\mathcal{N}=2$ supercurrent
multiplets in four dimensions. It is shown that associated with every
conformal supercurrent $J_{\alpha(m)\dot{\alpha}(n)}$ (with $m,n$ non-negative
integers) is a descendant $J^{ij}_{\alpha(m+1){\dot{\alpha}}(n+1)}$ with the
following properties: (a) it is a linear multiplet with respect to its
$\mathsf{SU}(2)$ indices, that is
$D_{\beta}^{(i}J^{jk)}_{\alpha(m+1)\dot{\alpha}(n+1)}=0$ and
$\bar{D}_{\dot{\beta}}^{(i}J^{jk)}_{\alpha(m+1)\dot{\alpha}(n+1)}=0$; and (b)
it is conserved,
$\partial^{\beta\dot{\beta}}J^{ij}_{\beta\alpha(m)\dot{\beta}\dot{\alpha}(n)}=0$.
Realisations of the conformal supercurrents $J_{\alpha(s)\dot{\alpha}(s)}$,
with $s=0,1,\dots$, are naturally provided by a massless hypermultiplet and a
vector multiplet. It turns out that such supercurrents and their linear
descendants $J^{ij}_{\alpha(s+1)\dot{\alpha}(s+1)}$ do not occur in the
harmonic-superspace framework recently described in arXiv:2212.14114. Making
use of a massive hypermultiplet, we derive non-conformal higher-spin
$\mathcal{N}=2$ supercurrent multiplets.

##  1 Introduction

Almost two years ago, we described all possible ${\cal N}=2$ conformal
supercurrent multiplets $J^{\alpha(m){\dot{\alpha}}(n)}$ in a curved
supergravity background [1], with $m$ and $n$ non-negative integers. The
$m=n=0$ case corresponds to the ordinary conformal supercurrent $J$ [2, 3],
which is the source of the Weyl multiplet of ${\cal N}=2$ conformal
supergravity [4]. The supergravity origin of $J$ and its non-conformal
extensions was uncovered in [5, 6]. If at least one of the parameters $m,n$ is
non-zero, the supercurrent $J^{\alpha(m){\dot{\alpha}}(n)}$ is the source of a
superconformal primary higher-spin gauge multiplet
$\Upsilon_{\alpha(m){\dot{\alpha}}(n)}$. The corresponding superconformal
gauge-invariant action for $\Upsilon_{\alpha(m){\dot{\alpha}}(n)}$ was
constructed in [1] in an arbitrary conformally flat background.

For a massless hypermultiplet, the conserved conformal supercurrent multiplets
$J^{\alpha(s){\dot{\alpha}}(s)}$, with $s\geq 0$, were derived in [1] in an
arbitrary conformally flat background. In the $s=0$ case, the conserved
massless hypermultiplet supercurrent $J$ exists in an arbitrary supergravity
background [5]. It should be stressed that each conserved conformal
hypermultiplet supercurrent $J^{\alpha(s){\dot{\alpha}}(s)}$ is uniquely
defined by the corresponding conservation equation. In particular, this
equation implies that $J^{\alpha(s){\dot{\alpha}}(s)}$ is a conformal primary
superfield.

For a massive hypermultiplet in ${\cal N}=2$ Minkowski superspace, conserved
supercurrent multiplets $J^{\alpha(s){\dot{\alpha}}(s)}$ can be derived from
the massless ones given in [1] by coupling the hypermultiplet to a ‘frozen’
vector multiplet à la [7].111In the $s=0$ case, the conserved massive
hypermultiplet supercurrent $J$ exists in an arbitrary supergravity background
[5]. The hypermultiplet’s mass is generated by coupling it to a vector
multiplet, one of the two supergravity compensators. Following [8], the
supercurrent conservation equation has the form $\frac{1}{4}(\bar{\cal
D}^{ij}+4\bar{S}^{ij})J={\cal W}T^{ij}$, where $\bar{\cal D}^{ij}=\bar{\cal
D}_{\dot{\alpha}}^{(i}\bar{\cal D}^{j){\dot{\alpha}}}$, $\bar{S}^{ij}$ is one
of the torsion tensors, ${\cal W}$ is the chiral field strength of the vector
multiplet and $T^{ij}$ is the supertrace obeying the constraints ${\cal
D}_{\alpha}^{(k}T^{ij)}=0$ and $\bar{\cal D}_{\dot{\alpha}}^{(k}T^{ij)}=0$. In
the massless case, $T^{ij}=0$.  This is achieved by replacing the flat
covariant derivatives
$D_{A}=(\partial_{a},D_{\alpha}^{i},\bar{D}^{{\dot{\alpha}}i})$ with gauge-
covariant ones ${\mbox{\boldmath$D$}}_{A}$, eq. (3.4), with the chiral field
strength being constant.

It should be pointed out that the conserved higher-spin supercurrent
multiplets for a free hypermultiplet have also been described in terms of
${\cal N}=1$ superfields. In the massless hypermultiplet case, the
corresponding higher-spin conformal supercurrents were derived in [9] in
${\cal N}=1$ Minkowski superspace. In ${\cal N}=1$ anti-de Sitter superspace,
the higher-spin hypermultiplet supercurrents were described in [10], both in
the massless and massive cases.

Recently, using the ${\cal N}=2$ harmonic superspace approach, conserved
hypermultiplet superfield currents have been presented by Buchbinder, Ivanov
and Zaigraev [11], building on their earlier construction of massless higher-
spin ${\cal N}=2$ supersymmetric gauge actions [12, 13]. No comparison with
the existing results on the hypermultiplet supercurrent multiplets [1, 9, 10]
was given in [11]. In our opinion, carrying out such a comparison is
important, and this short paper provides a thorough analysis.

There exists a duality between massless higher-spin (super)field actions and
non-conformal higher-spin (super)currents. In general, massless models are
formulated in terms of a uniquely defined conformal gauge (super)field in
conjunction with certain compensator (super)fields. In the non-supersymmetric
case, this is true for Fronsdal’s massless spin-$s$ field action [14], with
$s\geq 2$ an integer, which is formulated using two fields, the conformal
prepotential $h_{\alpha(s){\dot{\alpha}}(s)}$ and the compensator
$h_{\alpha(s-2){\dot{\alpha}}(s-2)}$. This is also true in the ${\cal N}=1$
supersymmetric case. For instance, there are two off-shell formulations for
the massless half-integer superspin multiplet [15] which involve the same
conformal prepotential $H_{\alpha(s){\dot{\alpha}}(s)}$, which is real and
unconstrained, but different compensators. They are described in terms of the
following dynamical variables:

$\displaystyle{\cal V}^{\bot}_{s+1/2}$ $\displaystyle=$
$\displaystyle\Big{\\{}H_{\alpha(s){\dot{\alpha}}(s)}~{},~{}\Gamma_{\alpha(s-1){\dot{\alpha}}(s-1)}~{},~{}\bar{\Gamma}_{\alpha(s-1){\dot{\alpha}}(s-1)}\Big{\\}}~{},$
(1.1a) $\displaystyle{\cal V}^{\|}_{s+1/2}$ $\displaystyle=$
$\displaystyle\Big{\\{}H_{\alpha(s){\dot{\alpha}}(s)}~{},~{}G_{\alpha(s-1){\dot{\alpha}}(s-1)}~{},~{}\bar{G}_{\alpha(s-1){\dot{\alpha}}(s-1)}\Big{\\}}~{}.$
(1.1b)

Here the complex superfields $\Gamma_{\alpha(s-1){\dot{\alpha}}(s-1)}$ and
$G_{\alpha(s-1){\dot{\alpha}}(s-1)}$ are transverse linear and longitudinal
linear, respectively,

$\displaystyle{\bar{D}}^{\dot{\beta}}\,\Gamma_{\alpha(s-1){\dot{\beta}}{\dot{\alpha}}(s-2)}$
$\displaystyle=$ $\displaystyle 0~{},$ (1.2a)
$\displaystyle{\bar{D}}_{({\dot{\alpha}}_{1}}\,G_{\alpha(s-1){\dot{\alpha}}_{2}\dots\dot{\alpha}_{s})}$
$\displaystyle=$ $\displaystyle 0~{}.$ (1.2b)

Different choices of conformal compensators lead to different non-conformal
conservation equations [10, 16]. In the ${\cal N}=2$ supersymmetric case, the
problem of classifying different off-shell massless higher-superspin models is
equivalent to that of classifying non-conformal higher-spin supercurrent
multiplets. We believe that the massless higher-spin ${\cal N}=2$
supersymmetric gauge models proposed in [12] are not unique. One way to
uncover other off-shell massless higher-superspin actions is to study
different non-conformal higher-spin supercurrent multiplets. This paper
initiates the program of classifying the non-conformal higher-spin ${\cal
N}=2$ supercurrent multiplets.

This paper is organised as follows. In section 2 we review the conformal
${\cal N}=2$ higher-spin supercurrents and corresponding multiplets in the
case of a massless hypermultiplet. Additionally, we construct conformal
higher-spin supercurrents for a vector multiplet. Section 3 is devoted to the
construction of non-conformal higher-spin supercurrents for a massive
hypermultiplet. The former are also compared with the supercurrents recently
proposed in [11]. The main body of this paper is accompanied by a technical
appendix, namely appendix A, reviewing conformal supercurrents in ${\cal N}=1$
superspace.

##  2 Conformal higher-spin supercurrents

The conformal ${\cal N}=2$ supercurrent multiplets
$J^{\alpha(m){\dot{\alpha}}(n)}$ were introduced in [1] in a curved
supergravity background. For our purposes it suffices to work in ${\cal N}=2$
Minkowski superspace where the algebra of covariant derivatives
$D_{A}=(\partial_{a},D_{\alpha}^{i},\bar{D}^{{\dot{\alpha}}i})$ is determined
by

$\displaystyle\\{D_{\alpha}^{i},\bar{D}^{\dot{\beta}}_{j}\\}=-2{\rm
i}\,\delta_{j}^{i}\partial_{\alpha}{}^{\dot{\beta}}~{},$ (2.1)

with all other graded commutation relations being trivial.

###  2.1 ${\cal N}=2$ conserved supercurrent multiplets

A conformal primary222See, e.g. [5, 17] for the definition of a primary tensor
superfield in flat superspace. tensor superfield
$J^{\alpha(m){\dot{\alpha}}(n)}$, with $m,n\geq 1$, is said to be a conformal
supercurrent if it obeys the constraints [1]

$\displaystyle D_{\beta}^{i}J^{\beta\alpha(m-1){\dot{\alpha}}(n)}$
$\displaystyle=$ $\displaystyle 0\quad\Longrightarrow\quad
D^{ij}J^{\alpha(m){\dot{\alpha}}(n)}=0~{},$ (2.2a)
$\displaystyle\bar{D}_{\dot{\beta}}^{i}J^{\alpha(m){\dot{\beta}}{\dot{\alpha}}(n-1)}$
$\displaystyle=$ $\displaystyle
0\quad\Longrightarrow\quad\bar{D}^{ij}J^{\alpha(m){\dot{\alpha}}(n)}=0~{}.$
(2.2b)

Here we have employed the shorthand

$\displaystyle{D}^{ij}={D}^{\alpha(i}\bar{D}_{\alpha}^{j)}~{},\qquad\bar{D}^{ij}=\bar{D}_{{\dot{\alpha}}}^{(i}\bar{D}^{{\dot{\alpha}}j)}~{}.$
(2.3)

In the special case $m=n=s$, the supercurrent $J^{\alpha(s){\dot{\alpha}}(s)}$
is restricted to be real. This case was first described in [3]. Setting $n=0$,
the conformal supercurrents $J^{\alpha(m)}$ instead satisfy

$\displaystyle D_{\beta}^{i}J^{\beta\alpha(m-1)}$ $\displaystyle=$
$\displaystyle 0\quad\Longrightarrow\quad D^{ij}J^{\alpha(m)}=0~{},$ (2.4a)
$\displaystyle\bar{D}^{ij}J^{\alpha(m)}$ $\displaystyle=$ $\displaystyle
0~{}.$ (2.4b)

Finally, when $m=n=0$, $J$ is real and corresponds to the conformal
supercurrent of [3, 2, 5]

$\displaystyle D^{ij}J=0~{},\quad\bar{D}^{ij}J=0~{}.$ (2.5)

The above constraints imply that, at the component level,
$J^{\alpha(m){\dot{\alpha}}(n)}$ contains a number of conserved currents. The
component analysis of $J^{\alpha(m){\dot{\alpha}}(n)}$ can be performed in two
stages. First, each supercurrent $J^{\alpha(m){\dot{\alpha}}(n)}$ can be
decomposed as a collection of conformal ${\cal N}=1$ supercurrent multiplets,
which is discussed in appendix A. Second, every conformal ${\cal N}=1$
supercurrent can be reduced to components to result with several ordinary
conformal currents. This stage is described in [18].

One can find a general local solution to the constraints (2.2), (2.4) and
(2.5) using the ${\cal N}=2$ superspin projection operators [19].

###  2.2 The hypermultiplet

An on-shell massless hypermultiplet is described by a conformal primary
isospinor $q^{i}$ (and its conjugate $\bar{q}_{i}$) subject to the constraints

$\displaystyle
D_{\alpha}^{(i}q^{j)}=0~{},\quad\bar{D}_{{\dot{\alpha}}}^{(i}q^{j)}=0~{},$
(2.6)

From $q^{i}$ and its conjugate, we construct the higher-derivative descendants
[1]

$\displaystyle J^{\alpha(s){\dot{\alpha}}(s)}$ $\displaystyle=$
$\displaystyle-\frac{{\rm i}^{s}}{2}\sum^{s}_{k=0}(-1)^{k}{s\choose
k}^{2}\partial^{(\alpha_{1}({\dot{\alpha}}_{1}}\dots\partial^{\alpha_{k}{\dot{\alpha}}_{k}}q^{i}\partial^{\alpha_{k+1}{\dot{\alpha}}_{k+1}}\dots\partial^{\alpha_{s}){\dot{\alpha}}_{s})}\bar{q}_{i}$
(2.7) $\displaystyle+\frac{{\rm i}^{s+1}}{16}\sum_{k=0}^{s-1}(-1)^{k}{s\choose
k}{s\choose k+1}$
$\displaystyle\qquad\times\bigg{\\{}\partial^{(\alpha_{1}({\dot{\alpha}}_{1}}\dots\partial^{\alpha_{k}{\dot{\alpha}}_{k}}D^{\alpha_{k+1}i}q_{i}\partial^{\alpha_{k+2}{\dot{\alpha}}_{k+1}}\dots\partial^{\alpha_{s}){\dot{\alpha}}_{s-1}}\bar{D}^{{\dot{\alpha}}_{s})j}\bar{q}_{j}$
$\displaystyle\qquad\quad-\partial^{(\alpha_{1}({\dot{\alpha}}_{1}}\dots\partial^{\alpha_{k}{\dot{\alpha}}_{k}}\bar{D}^{{\dot{\alpha}}_{k+1}i}q_{i}\partial^{\alpha_{k+1}{\dot{\alpha}}_{k+2}}\dots\partial^{\alpha_{s-1}{\dot{\alpha}}_{s})}D^{\alpha_{s})j}\bar{q}_{j}\bigg{\\}}~{}.$

It may be shown that the superfield (2.7) is conformally primary and satisfies
the conservation equations (2.2) for $s>0$ and (2.5) when $s=0$. Thus,
$J^{\alpha(s){\dot{\alpha}}(s)}$ is a conformal supercurrent. The $s=0$ case
was studied in a supergravity background in [5].

###  2.3 The vector multiplet

Next, we consider an on-shell vector multiplet, which is described by a
complex scalar $W$ (and its conjugate $\bar{W}$) subject to the constraints

$\displaystyle\bar{D}_{{\dot{\alpha}}}^{i}W=0~{},\quad D^{ij}W=0~{}.$ (2.8)

From $W$ and its conjugate, one may construct the higher-derivative
descendants333It should be noted that this supercurrent may be extended to
general conformally-flat backgrounds by the replacement
$D_{A}\rightarrow\nabla_{A}$, where $\nabla_{A}$ is the conformally covariant
derivative, see e.g. [20, 21].

$\displaystyle J^{\alpha(s){\dot{\alpha}}(s)}$ $\displaystyle=$
$\displaystyle{\rm i}^{s}\sum^{s}_{k=0}(-1)^{k}{s\choose
k}^{2}\partial^{(\alpha_{1}({\dot{\alpha}}_{1}}\dots\partial^{\alpha_{k}{\dot{\alpha}}_{k}}W\partial^{\alpha_{k+1}{\dot{\alpha}}_{k+1}}\dots\partial^{\alpha_{s}){\dot{\alpha}}_{s})}\bar{W}$
(2.9) $\displaystyle-$ $\displaystyle\frac{{\rm
i}^{s+1}}{2}\sum_{k=0}^{s-1}(-1)^{k}{s\choose k}{s\choose
k+1}\partial^{(\alpha_{1}({\dot{\alpha}}_{1}}\dots\partial^{\alpha_{k}{\dot{\alpha}}_{k}}D^{\alpha_{k+1}i}W$
$\displaystyle\qquad\qquad\times\partial^{\alpha_{k+2}{\dot{\alpha}}_{k+1}}\dots\partial^{\alpha_{s}){\dot{\alpha}}_{s-1}}\bar{D}^{{\dot{\alpha}}_{s})}_{i}\bar{W}$
$\displaystyle+$ $\displaystyle\frac{{\rm
i}^{s}}{16}\sum_{k=0}^{s-2}(-1)^{k}{s\choose k}{s\choose
k+2}\partial^{(\alpha_{1}({\dot{\alpha}}_{1}}\dots\partial^{\alpha_{k}{\dot{\alpha}}_{k}}D^{\alpha_{k+1}\alpha_{k+2}}W$
$\displaystyle\qquad\qquad\times\partial^{\alpha_{k+3}{\dot{\alpha}}_{k+1}}\dots\partial^{\alpha_{s}){\dot{\alpha}}_{s-2}}\bar{D}^{{\dot{\alpha}}_{s-1}{\dot{\alpha}}_{s})}\bar{W}~{},$

where we have made the definitions

$\displaystyle
D_{\alpha\beta}=D_{(\alpha}^{i}D_{\beta)i}~{},\qquad\bar{D}_{{\dot{\alpha}}{\dot{\beta}}}=\bar{D}_{({\dot{\alpha}}i}\bar{D}_{{\dot{\beta}})}^{i}~{}.$
(2.10)

The $s=0$ case was considered in [3]. It may be shown that (2.7) is primary
and satisfies the conservation equations (2.2) for $s>0$ and (2.5) when $s=0$,
hence it constitutes a conformal supercurrent.

###  2.4 Linear descendants

Given a conformal supercurrent $J_{\alpha(m){\dot{\alpha}}(n)}$, the
conservation equations (2.2), (2.4) and (2.5) imply the relation

$\displaystyle\big{[}D^{ij},\bar{D}^{kl}\big{]}J_{\alpha(m){\dot{\alpha}}(n)}=0~{}.$
(2.11)

Its use allows one to demonstrate that the following descendant of
$J^{\alpha(m){\dot{\alpha}}(n)}$,

$\displaystyle
J^{ij}_{\alpha(m+1){\dot{\alpha}}(n+1)}:=\frac{1}{2}\big{[}D_{(\alpha_{1}}^{(i},\bar{D}^{j)}_{({\dot{\alpha}}_{1}}\big{]}J_{\alpha_{2}\dots\alpha_{m+1}){\dot{\alpha}}_{2}\dots\dot{\alpha}_{n+1})}$
(2.12)

is Grassmann analytic,

$\displaystyle
D_{\beta}^{(i}J^{jk)}_{\alpha(m+1){\dot{\alpha}}(n+1)}=0~{},\qquad\bar{D}_{\dot{\beta}}^{(i}J^{jk)}_{\alpha(m+1){\dot{\alpha}}(n+1)}=0~{},$
(2.13a) and conserved,
$\displaystyle\partial^{\beta{\dot{\beta}}}J^{ij}_{\alpha(m)\beta{\dot{\alpha}}(n){\dot{\beta}}}=0~{}.$
(2.13b)

It is not difficult to check that, unlike $J_{\alpha(m){\dot{\alpha}}(n)}$,
the descendant $J^{ij}_{\alpha(m+1){\dot{\alpha}}(n+1)}$ is not a conformal
primary superfield. It follows from (2.13a) that
$J^{ij}_{\alpha(m+1){\dot{\alpha}}(n+1)}$ is a linear multiplet with respect
to its $\mathsf{SU}(2)$ indices. Using the projective-superspace terminology
[22, 23], $J^{ij}_{\alpha(m+1){\dot{\alpha}}(n+1)}$ can also be called an
${\cal O}(2)$ multiplet with respect to its $\mathsf{SU}(2)$ indices.

As is well known [3, 24]), given a linear multiplet $G^{ij}=G^{ji}$
constrained by $D_{\beta}^{(i}G^{jk)}=0$ and
$\bar{D}_{\dot{\beta}}^{(i}G^{jk)}=0$, it contains a transverse vector defined
by $V_{\beta{\dot{\beta}}}=\frac{{\rm
i}}{2}[D^{i}_{\beta},\bar{D}^{j}_{\dot{\beta}}]G_{ij}$ such that
$\partial^{\beta{\dot{\beta}}}V_{\beta{\dot{\beta}}}=0$. Therefore, associated
with the descendant (2.12) is the superfield

$\displaystyle
J_{\beta{\dot{\beta}},\alpha(m+1){\dot{\alpha}}(n+1)}:=\frac{{\rm
i}}{2}[D_{\beta
i},\bar{D}_{{\dot{\beta}}j}]J^{ij}_{\alpha(m+1){\dot{\alpha}}(n+1)}~{},$
(2.14)

which is separately transverse in its $\beta{\dot{\beta}}$ and
$\alpha{\dot{\alpha}}$ indices,

$\displaystyle\partial^{\beta{\dot{\beta}}}J_{\beta{\dot{\beta}},\alpha(m+1){\dot{\alpha}}(n+1)}$
$\displaystyle=$ $\displaystyle 0~{},$ (2.15a)
$\displaystyle\partial^{\gamma{\dot{\gamma}}}J_{\beta{\dot{\beta}},\gamma\alpha(m){\dot{\gamma}}{\dot{\alpha}}(n)}$
$\displaystyle=$ $\displaystyle 0~{}.$ (2.15b)

For comparison with the results of [11], it is convenient to start using the
harmonic superspace conventions [25] (see [26] for a review). We introduce
harmonics $u^{+}_{i}$ and $u^{-}_{i}$ parametrising the group
$\mathsf{SU}(2)$,

$\displaystyle\overline{u^{+i}}=u^{-}_{i}~{},\qquad u^{+i}u^{-}_{i}=1~{},$
(2.16)

and define a new basis for the spinor covariant derivatives,

$\displaystyle
D^{\pm}_{\alpha}=D^{i}_{\alpha}\,u^{\pm}_{i}~{},\qquad{\bar{D}}^{\pm}_{\dot{\alpha}}={\bar{D}}^{i}_{\dot{\alpha}}\,u^{\pm}_{i}~{},$
(2.17)

such that eq. (2.1) takes the form

$\displaystyle\\{{\bar{D}}^{+}_{\dot{\alpha}},D^{-}_{\alpha}\\}=-\\{D^{+}_{\alpha},{\bar{D}}^{-}_{\dot{\alpha}}\\}=2{\rm
i}\,\partial_{\alpha{\dot{\alpha}}}~{},$ (2.18)

The operators $D^{+}_{\alpha}$ and $\bar{D}^{+}_{\dot{\alpha}}$ anticommute
with each other. By definition, the analytic superfields
$\phi^{(n)}(z,u^{\pm})$ are annihilated by $D^{+}_{\alpha}$ and
$\bar{D}^{+}_{\dot{\alpha}}$. The superscript ‘$n$’ of $\phi^{(n)}$ denotes
the harmonic $\mathsf{U}(1)$ charge of $\phi^{(n)}$ defined by
$D^{0}\phi^{(n)}=n\phi^{(n)}$, where $D^{0}$ is one of the harmonic
derivatives

$\displaystyle D^{++}=u^{+i}\frac{\partial}{\partial u^{-i}}~{},\quad
D^{--}=u^{-i}\frac{\partial}{\partial u^{+i}}~{},\quad D^{0}$ $\displaystyle=$
$\displaystyle u^{+i}\frac{\partial}{\partial
u^{+i}}-u^{-i}\frac{\partial}{\partial u^{-i}}~{},$ (2.19)

which form a basis in the space of left-invariant vector fields on
$\mathsf{SU}(2)$.

Associated with the descendant (2.12) is the harmonic superfield

$\displaystyle
J^{++\alpha(m+1){\dot{\alpha}}(n+1)}=u^{+}_{i}u^{+}_{j}J^{ij\alpha(m+1){\dot{\alpha}}(n+1)}$
(2.20)

with the following properties:

$\displaystyle
D^{+}_{\beta}J^{++\alpha(m+1){\dot{\alpha}}(n+1)}=0~{},\qquad\bar{D}^{+}_{\dot{\beta}}J^{++\alpha(m+1){\dot{\alpha}}(n+1)}=0~{},$
(2.21a) $\displaystyle D^{++}J^{++\alpha(m+1){\dot{\alpha}}(n+1)}=0~{},$
(2.21b)
$\displaystyle\partial_{\beta{\dot{\beta}}}J^{++\alpha(m)\beta{\dot{\alpha}}(n){\dot{\beta}}}=0~{}.$
(2.21c)

In the case of the hypermultiplet supercurrent (2.7), its descendant
$J^{++\alpha(s+1){\dot{\alpha}}(s+1)}$ has the form:

$\displaystyle J^{++\alpha(s+1){\dot{\alpha}}(s+1)}={\rm
i}^{s+1}\sum_{k=0}^{s+1}(-1)^{k}{s+1\choose
k}^{2}\partial^{(\alpha_{1}({\dot{\alpha}}_{1}}\dots\partial^{\alpha_{k}{\dot{\alpha}}_{k}}q^{+}\partial^{\alpha_{k+1}{\dot{\alpha}}_{k+1}}\dots\partial^{\alpha_{s+1}){\dot{\alpha}}_{s+1})}\breve{q}^{+}~{},$
(2.22)

where $\breve{q}{}^{+}=-{\bar{q}}^{i}u^{+}_{i}$ is the smile-conjugate of
$q^{+}=q^{i}u^{+}_{i}$. It obeys the Grassmann analyticity constraints (2.21a)
since the on-shell hypermultiplet superfields $q^{+}$ and $\breve{q}^{+}$ are
Grassmann analytic. It obeys the constraint (2.21b) since both $q^{+}$ and
$\breve{q}^{+}$ are annihilated by $D^{++}$. Finally, the explicit dependence
of $J^{++\alpha(s+1){\dot{\alpha}}(s+1)}$ on spacetime derivatives guarantees
the fulfilment of (2.21c).

Now we reproduce the current superfield for the massless hypermultiplet given
in [11]

$\displaystyle J^{++\alpha(s+1){\dot{\alpha}}(s+1)}_{\rm BIZ}$
$\displaystyle=-\frac{{\rm
i}^{\frac{1-(-1)^{s}}{2}}}{2}\Big{\\{}q^{+}\partial^{(\alpha_{1}({\dot{\alpha}}_{1}}\dots\partial^{\alpha_{s+1}){\dot{\alpha}}_{s+1})}\breve{q}^{+}-(-1)^{s}\breve{q}^{+}\partial^{(\alpha_{1}({\dot{\alpha}}_{1}}\dots\partial^{\alpha_{s+1}){\dot{\alpha}}_{s+1})}{q}^{+}$
$\displaystyle\phantom{=}+\xi(1-(-1)^{s})\partial^{(\alpha_{1}({\dot{\alpha}}_{1}}\dots\partial^{\alpha_{s+1}){\dot{\alpha}}_{s+1})}(q^{+}\breve{q}^{+})\Big{\\}}~{},$
(2.23)

where $\xi$ is a real parameter. It satisfies only the equations (2.21a) and
(2.21b). Therefore, at the component level it contains a single conserved
current. It is defined to be the $\theta$-independent component of (2.14) and
satisfies only the conservation equation (2.15a).

Actually, there are $2\lfloor\frac{s+3}{2}\rfloor$ linearly independent real
superfields of the given tensor structure,
${\mathfrak{J}}^{++\alpha(s+1){\dot{\alpha}}(s+1)}$, that satisfy the same
equations (2.21a) and (2.21b). Their general form is

$\displaystyle{\mathfrak{J}}^{++\alpha(s+1){\dot{\alpha}}(s+1)}=\sum_{k=0}^{s+1}c_{k}\,\partial^{(\alpha_{1}({\dot{\alpha}}_{1}}\dots\partial^{\alpha_{k}{\dot{\alpha}}_{k}}q^{+}\partial^{\alpha_{k+1}{\dot{\alpha}}_{k+1}}\dots\partial^{\alpha_{s+1}){\dot{\alpha}}_{s+1})}\breve{q}^{+}=\breve{\mathfrak{J}}^{++\alpha(s+1){\dot{\alpha}}(s+1)}~{},$
(2.24)

with arbitrary coefficients $c_{k}$ modulo natural reality conditions. It is
natural to ask the question: What is special about the supercurrent (2.23)?

The supercurrent (2.23) was derived in [11] via the Noether procedure, which
involved making used of the rigid symmetries of the hypermultiplet introduced
in [13]. Without requiring the hypermultiplet transformation laws to be
compatible with the ${\cal N}=2$ superconformal symmetry, there is a huge
freedom in the structure of such rigid symmetries of the hypermultiplet and,
as a result, in the explicit form of cubic interaction vertices and associated
conserved currents. In Refs. [11, 13], the authors fixed a particular non-
conformal family of rigid symmetries of the hypermultiplet. Consequently, the
corresponding conserved currents (2.23) are non-primary, which is in contrast
to the multiplet (2.7).444We refer the reader to [27] for a study of the
Noether procedure for matter coupled to (super)conformal higher-spin fields
with manifest (super)conformal invariance. Other currents (2.24) correspond to
different choices of rigid symmetries of the hypermultiplet.

A general local solution to the equations (2.21) is given by (2.20) in which
$J^{ij\alpha(m+1){\dot{\alpha}}(n+1)}$ has the form

$\displaystyle J^{ij}_{\alpha(m+1){\dot{\alpha}}(n+1)}$ $\displaystyle=$
$\displaystyle
D^{ij}\Phi_{\alpha(m+1){\dot{\alpha}}(n+1)}+\bar{D}^{ij}\bar{\Psi}_{\alpha(m+1){\dot{\alpha}}(n+1)}~{},$
(2.25a)
$\displaystyle\bar{D}_{j}^{\dot{\beta}}\Phi_{\alpha(m+1){\dot{\alpha}}(n+1)}$
$\displaystyle=$ $\displaystyle
0~{},\qquad\bar{D}_{j}^{\dot{\beta}}\Psi_{\alpha(n+1){\dot{\alpha}}(m+1)}=0~{},$
(2.25b) $\displaystyle\Phi_{\alpha(m+1){\dot{\alpha}}(n+1)}$ $\displaystyle=$
$\displaystyle\partial^{\beta_{1}}{}_{{\dot{\alpha}}_{1}}\dots\partial^{\beta_{n+1}}{}_{{\dot{\alpha}}_{n+1}}\partial_{(\beta_{1}}{}^{{\dot{\beta}}_{1}}\dots\partial_{\beta_{n+1}}{}^{{\dot{\beta}}_{n+1}}\phi_{\alpha_{1}\dots\alpha_{m+1}){\dot{\beta}}_{1}\dots\dot{\beta}_{n+1}}~{},$
(2.25c) $\displaystyle\Psi_{\alpha(n+1){\dot{\alpha}}(m+1)}$ $\displaystyle=$
$\displaystyle\partial^{\beta_{1}}{}_{{\dot{\alpha}}_{1}}\dots\partial^{\beta_{m+1}}{}_{{\dot{\alpha}}_{m+1}}\partial_{(\beta_{1}}{}^{{\dot{\beta}}_{1}}\dots\partial_{\beta_{m+1}}{}^{{\dot{\beta}}_{m+1}}\psi_{\alpha_{1}\dots\alpha_{n+1}){\dot{\beta}}_{1}\dots\dot{\beta}_{m+1}}~{},$
(2.25d)

with $\phi_{\alpha(m+1){\dot{\alpha}}(n+1)}$ and
$\psi_{\alpha(n+1){\dot{\alpha}}(m+1)}$ being arbitrary chiral superfields. In
the $m=n$ case, these chiral superfields coincide. The relations (2.25c) and
(2.25d) are not included if only the constraints (2.21a) and (2.21b) are
imposed, as in the case of (2.23).

##  3 Non-conformal higher-spin supercurrents

This section is devoted to an analysis of non-conformal higher-spin
supercurrents for the massive hypermultiplet. Additionally, by making use of
the results of the previous subsection, the modified conservation law of the
analytic descendant (2.22) is derived.

Before turning to the higher-spin story, it is worth recalling the explicit
structure of the non-conformal supercurrent [8] in the ${\cal N}=2$
supergravity formulation of [28]. This supergravity formulation makes use of
two compensators: the vector multiplet and the tensor multiplet. In the
superspace setting, the Weyl multiplet is described in terms of the conformal
prepotential ${\cal H}$, which is real and unconstrained. The vector multiplet
is described using Mezincescu’s prepotential ${\cal V}_{ij}$, which is a real
iso-triplet. The prepotentials for the tensor multiplet are a chiral scalar
$\Psi$ and its conjugate. The supercurrent conservation equation, which
corresponds to a matter model with action $S$, was derived in [8] and has the
form (with ${\cal W}$ and ${\cal G}^{ij}$ being the field strengths of the
vector and tensor compensators, respectively)

$\displaystyle\frac{1}{4}(\bar{\cal D}^{ij}+4\bar{S}^{ij}){\cal J}={\cal
W}{\cal T}^{ij}-{\cal G}^{ij}{\cal Y}~{},$ (3.1)

where ${\cal J}$ denotes the supercurrent, while ${\cal T}^{ij}$ and ${\cal
Y}$ are the trace multiplets.

$\displaystyle{\cal J}=\frac{{\delta}S}{{\delta}{\cal H}}~{},\qquad{\cal
T}^{ij}=\frac{{\delta}S}{{\delta}{\cal V}_{ij}}~{},\qquad{\cal
Y}=\frac{{\delta}S}{{\delta}\Psi}~{}.$ (3.2)

The multiplet ${\cal J}$ and ${\cal T}_{ij}$ must be real, and ${\cal Y}$
covariantly chiral. In addition, ${\cal Y}$ and ${\cal T}_{ij}$ must obey the
constraints

$\displaystyle{\cal D}_{\alpha}{}^{(k}{\cal T}^{ij)}=\bar{\cal
D}_{\dot{\alpha}}{}^{(k}{\cal T}^{ij)}=0~{},$ (3.3a) $\displaystyle({\cal
D}^{ij}+4S^{ij}){\cal Y}=(\bar{\cal D}^{ij}+4\bar{S}^{ij})\bar{\cal Y}~{}.$
(3.3b)

In the higher-spin case, we know the analogue of ${\cal H}$, which is the
superconformal prepotential $H_{\alpha(s){\dot{\alpha}}(s)}$, though we do not
know yet the higher-spin analogues of the compensators ${\cal V}_{ij}$ and
$\Psi$. We hope to learn about their structure by analysing non-conformal
higher-spin supercurrents.

###  3.1 The massive hypermultiplet

In flat superspace, it is possible to endow a hypermultiplet with a mass by
coupling it to a ‘frozen’ vector multiplet à la [7]. This approach is most
useful to deal with off-shell hypermultiplets without intrinsic central
charge, such as the $q^{+}$ hypermultiplet [25] and the polar hypermultiplet
(see [22, 23] and references therein). It can also be used for the Fayet-
Sohnius hypermultiplet [29, 30] which suffices for our goals. To this end, we
introduce the gauge covariant derivatives

$\displaystyle\mbox{\boldmath$D$}_{A}=(\mbox{\boldmath$D$}_{a},\mbox{\boldmath$D$}_{\alpha}^{i},\bar{\mbox{\boldmath$D$}}^{{\dot{\alpha}}}_{i})=D_{A}+V_{A}\Delta~{},$
(3.4)

where $\Delta$ is a central charge operator,
$[\Delta,\mbox{\boldmath$D$}_{A}]=0$. The covariant derivatives
$\mbox{\boldmath$D$}_{A}$ are characterised by the anti-commutation relations

$\displaystyle\\{\mbox{\boldmath$D$}_{\alpha}^{i},\mbox{\boldmath$D$}_{\beta}^{j}\\}=-2\varepsilon^{ij}\varepsilon_{\alpha\beta}\bar{\mbox{\boldmath$W$}}_{0}\Delta~{},\quad\\{{\bar{\mbox{\boldmath$D$}}}_{\dot{\alpha}i}\,,{\bar{\mbox{\boldmath$D$}}}_{\dot{\beta}j}\\}=-2\varepsilon_{ij}\,\varepsilon_{\dot{\alpha}\dot{\beta}}{\mbox{\boldmath$W$}}_{0}\Delta~{},$
(3.5a)
$\displaystyle\\{\mbox{\boldmath$D$}_{\alpha}^{i},\bar{\mbox{\boldmath$D$}}^{\dot{\beta}}_{j}\\}=-2{\rm
i}\delta_{j}^{i}\mbox{\boldmath$D$}_{\alpha}{}^{\dot{\beta}}~{},$ (3.5b)

where $\mbox{\boldmath$W$}_{0}$ is a constant non-zero parameter. The gauge
freedom may be used to bring the gauge-covariant derivatives to the form

$\displaystyle{\mbox{\boldmath$D$}}_{\alpha}^{i}$ $\displaystyle=$
$\displaystyle\phantom{-}\frac{\partial}{\partial\theta^{\alpha}_{i}}+{\rm
i}\,(\sigma^{b})_{\alpha{\dot{\beta}}}\,{\bar{\theta}}^{\dot{\beta}i}\,\partial_{b}-\theta^{i}_{\alpha}\,\bar{\mbox{\boldmath$W$}}_{0}\Delta=D^{i}_{\alpha}-\theta^{i}_{\alpha}\,\bar{\mbox{\boldmath$W$}}_{0}\Delta~{},$
(3.6a) $\displaystyle{\bar{\mbox{\boldmath$D$}}}_{\dot{\alpha}i}$
$\displaystyle=$
$\displaystyle-\frac{\partial}{\partial{\bar{\theta}}^{\dot{\alpha}i}}-{\rm
i}\,\theta^{\beta}_{i}(\sigma^{b})_{\beta\dot{\alpha}}\,\partial_{b}+{\bar{\theta}}_{\dot{\alpha}i}\,{\mbox{\boldmath$W$}}_{0}\Delta={\bar{D}}_{{\dot{\alpha}}i}+{\bar{\theta}}_{\dot{\alpha}i}\,{\mbox{\boldmath$W$}}_{0}\Delta~{}.~{}~{}~{}$
(3.6b)

Without loss of generality, it is possible to choose

$\mbox{\boldmath$W$}_{0}={\rm i}~{}.$ (3.7)

To describe a massive hypermultiplet, the constraints (2.6) should be replaced
with

$\mbox{\boldmath$D$}_{\alpha}^{(i}q^{j)}=0~{},\quad\bar{\mbox{\boldmath$D$}}_{{\dot{\alpha}}}^{(i}q^{j)}=0~{}.$
(3.8)

An important consequence of (3.8) is

$\displaystyle(\mbox{\boldmath$D$}^{a}\mbox{\boldmath$D$}_{a}+\Delta^{2}){q}^{i}=0~{}.$
(3.9)

This is just a different way of looking at the Fayet-Sohnius hypermultiplet
[29, 30]. Thus, requiring that $q^{i}$ is an eigenvector of $\Delta$ (with
non-zero eigenvalue)

$\displaystyle\Delta q^{i}={\rm i}mq^{i}~{},\qquad
m\in\mathbb{R}\setminus\\{0\\}~{},$ (3.10)

is equivalent to endowing the hypermultiplet with a mass,

$\displaystyle(\mbox{\boldmath$D$}^{a}\mbox{\boldmath$D$}_{a}-m^{2})q^{i}=0~{}.$
(3.11)

Indeed, in the realisation (3.6) we have
${\mbox{\boldmath$D$}}_{a}=\partial_{a}$.

Having formulated the massive hypermultiplet above, its higher-spin
supercurrents are derived from (2.7) by performing the replacement
$D_{A}\rightarrow\mbox{\boldmath$D$}_{A}$. They satisfy the conservation
equations

$\displaystyle s=0:$ $\displaystyle\qquad{D}^{ij}J={\rm i}\mathbb{T}^{ij}~{},$
(3.12a) $\displaystyle s>0:$
$\displaystyle\qquad{D}_{\beta}^{i}J^{\alpha(s-1)\beta{\dot{\alpha}}(s)}={\rm
i}\bar{{D}}^{({\dot{\alpha}}_{1}}_{j}T^{ij\,\alpha(s-1){\dot{\alpha}}_{2}\dots\dot{\alpha}_{s})}+\bar{{D}}^{({\dot{\alpha}}_{1}i}S^{\alpha(s-1){\dot{\alpha}}_{2}\dots\dot{\alpha}_{s})}~{},$
(3.12b)

where we have introduced the real supertrace multiplets555The supertrace
$\mathbb{T}^{ij}$ was first derived in [5].

$\displaystyle\mathbb{T}^{ij}$ $\displaystyle=4m{\rm i}q^{(i}\bar{q}^{j)}~{},$
(3.13a) $\displaystyle T_{\alpha(s-1){\dot{\alpha}}(s-1)}^{ij}$
$\displaystyle=m{\rm
i}^{s}\sum_{k=0}^{s-1}\frac{(-1)^{k+1}(2k-s+1)s(s+1)(s+2)}{6(k+1)(k+2)(s-k+1)(s-k)}{s-1\choose
k}^{2}$
$\displaystyle\qquad\qquad\times\mbox{\boldmath$D$}_{(\alpha_{1}({\dot{\alpha}}_{1}}\dots\mbox{\boldmath$D$}_{\alpha_{k}{\dot{\alpha}}_{k}}q^{(i}\mbox{\boldmath$D$}_{\alpha_{k+1}{\dot{\alpha}}_{k+1}}\dots\mbox{\boldmath$D$}_{\alpha_{s-1}){\dot{\alpha}}_{s-1})}\bar{q}^{j)}~{},$
(3.13b) $\displaystyle S_{\alpha(s-1){\dot{\alpha}}(s-1)}$
$\displaystyle=m{\rm
i}^{s-1}\sum_{k=0}^{s-1}\frac{(-1)^{k+1}s(s+1)(s+2)(s+3)}{4(k+1)(k+2)(s-k+1)(s-k)}{s-1\choose
k}^{2}$
$\displaystyle\qquad\qquad\times\mbox{\boldmath$D$}_{(\alpha_{1}({\dot{\alpha}}_{1}}\dots\mbox{\boldmath$D$}_{\alpha_{k}{\dot{\alpha}}_{k}}q^{i}\mbox{\boldmath$D$}_{\alpha_{k+1}{\dot{\alpha}}_{k+1}}\dots\mbox{\boldmath$D$}_{\alpha_{s-1}){\dot{\alpha}}_{s-1})}\bar{q}_{i}~{}.$
(3.13c)

These multiplets are neutral with respect to the central charge. They satisfy
the following differential constraints:

$\displaystyle{D}_{\beta}^{(i}\mathbb{T}^{jk)}$
$\displaystyle=0~{},\qquad\bar{{D}}_{\dot{\beta}}^{(i}\mathbb{T}^{jk)}=0~{},$
(3.14a)
$\displaystyle{D}_{\beta}^{(i}T_{\alpha(s-1){\dot{\alpha}}(s-1)}^{jk)}$
$\displaystyle=0~{},\qquad\bar{{D}}_{\dot{\beta}}^{(i}T_{\alpha(s-1){\dot{\alpha}}(s-1)}^{jk)}=0~{},$
(3.14b) $\displaystyle{D}^{ij}S_{\alpha(s-1){\dot{\alpha}}(s-1)}$
$\displaystyle=\bar{{D}}^{ij}S_{\alpha(s-1){\dot{\alpha}}(s-1)}~{}.$ (3.14c)

The latter implies that

$\displaystyle
S^{ij}_{\alpha(s-1){\dot{\alpha}}(s-1)}:=\frac{1}{4}D^{ij}S_{\alpha(s-1){\dot{\alpha}}(s-1)}=\frac{1}{4}\bar{D}^{ij}S_{\alpha(s-1){\dot{\alpha}}(s-1)}$
(3.15)

is a linear multiplet with respect to its $\mathsf{SU}(2)$ indices

$\displaystyle{D}_{\beta}^{(i}S_{\alpha(s-1){\dot{\alpha}}(s-1)}^{jk)}$
$\displaystyle=0~{},\qquad\bar{{D}}_{\dot{\beta}}^{(i}S_{\alpha(s-1){\dot{\alpha}}(s-1)}^{jk)}=0~{}.$
(3.16)

It should be emphasised that there might be a different functional form for
the right-hand side of (3.12b). This will be discussed elsewhere.

###  3.2 Linear descendants

Let us analyse the massive hypermultiplet supercurrents introduced in the
previous subsection. As in the massless case, we can introduce the descendant
$J_{ij}^{\alpha(s+1){\dot{\alpha}}(s+1)}$, which is defined by the rule
(2.12), and the associated harmonic superfield
$J^{++\alpha(s+1){\dot{\alpha}}(s+1)}$, eq. (2.20).

$\displaystyle J^{++\alpha(s+1){\dot{\alpha}}(s+1)}={\rm
i}^{s+1}\sum_{k=0}^{s+1}(-1)^{k}{s+1\choose
k}^{2}\mbox{\boldmath$D$}^{(\alpha_{1}({\dot{\alpha}}_{1}}\dots\mbox{\boldmath$D$}^{\alpha_{k}{\dot{\alpha}}_{k}}q^{+}\mbox{\boldmath$D$}^{\alpha_{k+1}{\dot{\alpha}}_{k+1}}\dots\mbox{\boldmath$D$}^{\alpha_{s+1}){\dot{\alpha}}_{s+1})}\breve{q}^{+}~{},$
(3.17)

which is obtained from (2.22) upon the replacement
$\partial_{a}\rightarrow\mbox{\boldmath$D$}_{a}$. Making use of the
conservation equations (3.12) and (3.14), it is seen that
$J^{++\alpha(s+1){\dot{\alpha}}(s+1)}$ still obeys the constraints (2.21a) and
(2.21b). However, the constraint (2.21c) does not hold anymore. It is replaced
with

$\displaystyle\partial_{{\beta{\dot{\beta}}}}J^{++\alpha(s)\beta{\dot{\alpha}}(s){\dot{\beta}}}=\frac{s}{s+1}\partial^{(\alpha_{1}({\dot{\alpha}}_{1}}S^{++\alpha_{2}\dots\alpha{s}){\dot{\alpha}}_{2}\dots\dot{\alpha}_{s})}~{}.$
(3.18)

The relations (3.14a) and (3.14b) tell us that the harmonic superfields
$\mathbb{T}^{++}:=u^{+}_{i}u^{+}_{j}\mathbb{T}^{ij}$ and
$T^{++\alpha(s-1){\dot{\alpha}}(s-1)}:=u^{+}_{i}u^{+}_{j}T^{\alpha(s-1){\dot{\alpha}}(s-1)ij}$
are analytic. For $s>0$ the trace multiplets
$S^{\alpha(s-1){\dot{\alpha}}(s-1)}$ and
$T^{++\alpha(s-1){\dot{\alpha}}(s-1)}$ are related to each other as follows:

$\displaystyle
D^{+}_{\beta}\bar{D}^{+({\dot{\alpha}}_{1}}S^{\alpha(s-2)\beta{\dot{\alpha}}_{2}\dots\dot{\alpha}_{s-1})}+3\partial_{\beta}{}^{({\dot{\alpha}}_{1}}T^{++\alpha(s-2)\beta{\dot{\alpha}}_{2}\dots\dot{\alpha}_{s-1})}=0~{}.$
(3.19)

These results should also be compared with those for the massive
hypermultiplet given in [11]. In particular, we expect that they belong to the
family of superfields obtained from (2.24) upon the replacement
$\partial_{a}\rightarrow\mbox{\boldmath$D$}_{a}$

$\displaystyle{\mathfrak{J}}^{++\alpha(s+1){\dot{\alpha}}(s+1)}=\sum_{k=0}^{s+1}c_{k}\,\mbox{\boldmath$D$}^{(\alpha_{1}({\dot{\alpha}}_{1}}\dots\mbox{\boldmath$D$}^{\alpha_{k}{\dot{\alpha}}_{k}}q^{+}\mbox{\boldmath$D$}^{\alpha_{k+1}{\dot{\alpha}}_{k+1}}\dots\mbox{\boldmath$D$}^{\alpha_{s+1}){\dot{\alpha}}_{s+1})}\breve{q}^{+}=\breve{\mathfrak{J}}^{++\alpha(s+1){\dot{\alpha}}(s+1)}~{}.$
(3.20)

It is clear that these multiplets obey the constraints (2.21a) and (2.21b),
though they do not satisfy an analogue of (3.18). Hence, it is not clear what
is special about the non-conformal supercurrents employed in [11].

Acknowledgements: We are grateful to Michael Ponds for useful discussions. The
work of SK is supported in part by the Australian Research Council, project
No. DP200101944. The work of ER is supported by the Hackett Postgraduate
Scholarship UWA, under the Australian Government Research Training Program.

##  Appendix A Conformal supercurrents in ${\cal N}=1$ superspace

This appendix is aimed at reviewing the structure of conformal supercurrents
in ${\cal N}=1$ superspace. As shown below, they naturally arise from the
${\cal N}=2$ multiplets of section 2.1 upon performing a superspace reduction.

First, we sketch the procedure of reducing from ${\cal N}=2$ to ${\cal N}=1$
Minkowski superspace. Let $D_{\alpha}$, $\bar{D}^{\dot{\alpha}}$ and
$\partial_{{\alpha{\dot{\alpha}}}}=\frac{{\rm
i}}{2}\\{D_{\alpha},\bar{D}_{\dot{\alpha}}\\}$ be the covariant derivatives of
$\mathbb{M}^{4|4}$. They may be defined via the ${\cal N}=2$ covariant
derivatives as follows:
$D_{\alpha}{\mathfrak{U}}=D_{\alpha}^{\underline{1}}U|$ and
$\bar{D}^{\dot{\alpha}}{\mathfrak{U}}=\bar{D}^{{\dot{\alpha}}}_{\underline{1}}U|$.
Here $U$ is an ${\cal N}=2$ superfield, and ${\mathfrak{U}}\equiv
U|:=U|_{\theta^{\alpha}_{\underline{2}}=\bar{\theta}_{\dot{\alpha}}^{\underline{2}}=0}$
is its ${\cal N}=1$ projection.

We now review the basic properties of ${\cal N}=1$ conserved current
supermultiplets666We refer the reader to [18] for a more in-depth analysis.. A
primary tensor superfield $\mathcal{J}^{\alpha(m){\dot{\alpha}}(n)}$, with
$m,n\geq 1$, subject to the constraints

$\displaystyle D_{\beta}\mathcal{J}^{\beta\alpha(m-1){\dot{\alpha}}(n)}$
$\displaystyle=$ $\displaystyle 0\quad\implies\quad
D^{2}\mathcal{J}^{\alpha(m){\dot{\alpha}}(n)}=0~{},$ (A.1a)
$\displaystyle\bar{D}_{{\dot{\beta}}}\mathcal{J}^{\alpha(m){\dot{\beta}}{\dot{\alpha}}(n-1)}$
$\displaystyle=$ $\displaystyle
0\quad\implies\quad\bar{D}^{2}\mathcal{J}^{\alpha(m){\dot{\alpha}}(n)}=0~{},$
(A.1b)

is a conformal supercurrent. The $m=n=1$ case corresponds to the ordinary
conformal supercurrent [31]. For case $m>n=0$, (A.1) should be replaced with

$\displaystyle D_{\beta}\mathcal{J}^{\beta\alpha(m-1)}$ $\displaystyle=$
$\displaystyle 0\quad\implies\quad D^{2}\mathcal{J}^{\alpha(m)}=0~{},$ (A.2a)
$\displaystyle\bar{D}^{2}\mathcal{J}^{\alpha(m)}$ $\displaystyle=$
$\displaystyle 0~{}.$ (A.2b)

The $m=1$ case was first considered in [5], where it was shown that the spinor
supercurrent ${\cal J}^{\alpha}$ naturally originates from the reduction of
the conformal ${\cal N}=2$ supercurrent [2] to ${\cal N}=1$ superspace.
Finally, when $m=0$ the supercurrent satisfies

$\displaystyle D^{2}\mathcal{J}=0~{},\quad\bar{D}^{2}\mathcal{J}=0~{}.$ (A.3)

This is the flavour current supermultiplet [32]. The above ${\cal N}=1$
supercurrents each lead to a multiplet of conserved currents at the component
level, see [18] for a detailed study.

We first consider the ${\cal N}=2$ supercurrent
$J^{\alpha(m){\dot{\alpha}}(n)}$ (2.2). It contains four independent ${\cal
N}=1$ conserved current supermultiplets

$\displaystyle j^{\alpha(m){\dot{\alpha}}(n)}$ $\displaystyle=$ $\displaystyle
J^{\alpha(m){\dot{\alpha}}(n)}|~{},$ (A.4a) $\displaystyle
j^{\alpha(m+1){\dot{\alpha}}(n)}$ $\displaystyle=$ $\displaystyle
D^{(\alpha_{1}\underline{2}}J^{\alpha_{2}\dots\alpha_{m+1}){\dot{\alpha}}(n)}|~{},$
(A.4b) $\displaystyle j^{\alpha(m){\dot{\alpha}}(n+1)}$ $\displaystyle=$
$\displaystyle\bar{D}^{({\dot{\alpha}}_{1}}_{\underline{2}}J^{\alpha(m){\dot{\alpha}}_{2}\dots\dot{\alpha}_{n+1})}|~{},$
(A.4c) $\displaystyle j^{\alpha(m+1){\dot{\alpha}}(n+1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{[}D^{(\alpha_{1}\underline{2}},\bar{D}_{\underline{2}}^{({\dot{\alpha}}_{1}}\big{]}J^{\alpha_{2}\dots\alpha_{m+1}){\dot{\alpha}}_{2}\dots\dot{\alpha}_{n+1})}|$
(A.4d) $\displaystyle-$
$\displaystyle\frac{1}{2(m+n+3)}\big{[}D^{(\alpha_{1}},\bar{D}^{({\dot{\alpha}}_{1}}\big{]}j^{\alpha_{2}\dots\alpha_{m+1}){\dot{\alpha}}_{2}\dots\dot{\alpha}_{n+1})}$
$\displaystyle-$ $\displaystyle\frac{{\rm
i}(m-n)}{m+n+3}D^{(\alpha_{1}({\dot{\alpha}}_{1}}j^{\alpha_{2}\dots\alpha_{m+1}){\dot{\alpha}}_{2}\dots\dot{\alpha}_{n+1})}~{}.$

Similarly, $J^{\alpha(m)}$ (2.4) is composed of four ${\cal N}=1$
supercurrents

$\displaystyle j^{\alpha(m)}$ $\displaystyle=$ $\displaystyle
J^{\alpha(m)}|~{},$ (A.5a) $\displaystyle j^{\alpha(m+1)}$ $\displaystyle=$
$\displaystyle
D^{(\alpha_{1}\underline{2}}J^{\alpha_{2}\dots\alpha_{m+1})}|~{},$ (A.5b)
$\displaystyle j^{\alpha(m){\dot{\alpha}}}$ $\displaystyle=$
$\displaystyle\bar{D}^{{\dot{\alpha}}}_{\underline{2}}J^{\alpha(m)}|~{},$
(A.5c) $\displaystyle j^{\alpha(m+1){\dot{\alpha}}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{[}D^{(\alpha_{1}\underline{2}},\bar{D}_{\underline{2}}^{{\dot{\alpha}}}\big{]}J^{\alpha_{2}\dots\alpha_{m+1})}|-\frac{1}{2(m+3)}\big{[}D^{(\alpha_{1}},\bar{D}^{{\dot{\alpha}}}\big{]}j^{\alpha_{2}\dots\alpha_{m+1})}$
(A.5d) $\displaystyle-$ $\displaystyle\frac{{\rm
i}m}{m+3}\partial^{(\alpha_{1}{\dot{\alpha}}}j^{\alpha_{2}\dots\alpha_{m+1})}~{}.$

Finally, upon reduction of $J$ (2.5) we obtain three ${\cal N}=1$ current
multiplets [5]

$\displaystyle j$ $\displaystyle=$ $\displaystyle J|~{},$ (A.6a)
$\displaystyle j^{\alpha}$ $\displaystyle=$ $\displaystyle
D^{\alpha\underline{2}}J|~{},$ (A.6b) $\displaystyle j^{\alpha{\dot{\alpha}}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\big{[}D^{\alpha\underline{2}},\bar{D}_{\underline{2}}^{{\dot{\alpha}}}\big{]}J|-\frac{1}{6}\big{[}D^{\alpha},\bar{D}^{{\dot{\alpha}}}\big{]}j~{}.$
(A.6c)

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# Topological transitions in dissipatively coupled Su-Schrieffer-Heeger models

Jayakrishnan M. P. Nair<EMAIL_ADDRESS>Institute for Quantum
Science and Engineering, Texas A$\&$M University, College Station, TX 77843,
USA  Department of Physics and Astronomy, Texas A$\&$M University, College
Station, TX 77843, USA  Marlan O. Scully<EMAIL_ADDRESS>Institute for
Quantum Science and Engineering, Texas A$\&$M University, College Station, TX
77843, USA  Department of Physics and Astronomy, Texas A$\&$M University,
College Station, TX 77843, USA  Baylor University, Waco, TX 76704, USA
Princeton University, Princeton, New Jersey 08544, USA  Girish S. Agarwal
<EMAIL_ADDRESS>Institute for Quantum Science and Engineering,
Texas A$\&$M University, College Station, TX 77843, USA  Department of Physics
and Astronomy, Texas A$\&$M University, College Station, TX 77843, USA

###### Abstract

Non-Hermitian topological phenomena have gained much interest among physicists
in recent years. In this paper, we expound on the physics of dissipatively
coupled Su-Schrieffer-Heeger (SSH) lattices, specifically in systems with
bosonic and electrical constituents. In the context of electrical circuits, we
demonstrate that a series of resistively coupled LCR circuits mimics the
topology of a dissipatively coupled SSH model. In addition, we foreground a
scheme to construct dissipatively coupled SSH lattices involving a set of non-
interacting bosonic oscillators weakly coupled to engineered reservoirs of
modes possessing substantially small lifetimes when compared to other system
timescales. Further, by activating the coherent coupling between bosonic
oscillators, we elucidate the emergence of non-reciprocal dissipative coupling
which can be controlled by the phase of the coherent interaction strength
precipitating in phase-dependent topological transitions and skin effect. Our
analyses are generic, apropos of a large class of systems involving, for
instance, optical and microwave settings, while the circuit implementation
represents the most straightforward of them.

##  I Introduction

One of the prime objectives of research in condensed matter physics is the
characterisation of matter phases earmarked by the (spontaneous breaking of)
symmetries of the system under consideration. In this context, the discovery
of the quantum Hall effect marked a stark shift in the understanding of phases
and introduced the concept of topological order, spawning the field of
topological insulators [1, 2, 3]. This was subsequently realized on a variety
of different platforms, including, photonic [4], cold atomic systems [5] and
many more [6, 7, 8]. One of the key features of the topological classification
of phases is the bulk boundary correspondence (BBC) and the emergence of edge
and surface states that are impervious to environmental loss and disorder with
applications ranging from the realization of topological qubits [9, 10, 11,
12, 13] to lasing [14, 15, 16, 17] among others [18, 19, 20, 21, 22, 23, 24,
25].

Until recently, the lion’s share of research on the physics of topological
systems involved Hermitian models. However, real physical systems interact
with their environment resulting in open quantum dynamics [26] and effective
non-Hermitian Hamiltonians. In the last few years, the topology of non-
Hermitian lattice systems has been a subject of intense research activity [27,
28], unravelling some exciting new physics, for example, the breakdown of BBC
and skin effect in systems possessing non-reciprocal couplings [29, 30, 31,
32, 33, 34, 35, 36, 37]. A quintessential model in the study of topological
physics is the Su-Schrieffer-Heeger (SSH) model [38, 39, 40, 41, 42] and
several non-Hermitian extensions of the model have been considered in
literature [43, 44, 45, 46, 47]. For instance, [45, 46] considered
$PT$-symmetric extensions of the SSH model which can be engineered by the
incorporation gain into the system. However, the physics of lattice models
with a purely dissipative form of coupling [48, 49] between the constituents
is largely unplumbed. Dissipative coupling between two otherwise non-
interacting systems emanates from their decay into common dissipative channels
[50] and it is worth noting that dissipative couplings are more prevalent in
nature compared to their coherent counterparts. Such couplings have been
investigated both theoretically and experimentally in a multitude of settings
[51, 52, 53, 54, 55], for example, involving magnonic and photonic sub-systems
[56, 57, 58, 59, 60].

In this work, we focus on the physics of dissipatively coupled SSH models. In
particular, we demonstrate two distinct experimentally realizable schemes
involving bosonic and electrical subsystems. We show that a system of
resistively coupled LCR resonators mimics the topology of dissipatively
coupled SSH (DSSH) models. Subsequently, we illustrate that a lattice of
otherwise non-interacting bosonic oscillators interacting with engineered
reservoirs of modes having significantly large decay parameters compared to
other system parameters can be described by an effective non-Hermitian
Hamiltonian akin to the DSSH model. Furthermore, by triggering the coherent
coupling between the oscillators, we outline the generation of non-reciprocal
coupling in DSSH models featuring topological transitions that can be
controlled by the phase of the coherent interaction strength. Note, en
passant, the generality of our results grants an immediate experimental
realization of the protocols discussed in the subsequent sections, especially
in the microwave and optical domains.

This paper is organized as follows. In section II, we revisit the SSH model
with coherent couplings followed by a discussion of its dissipative
counterpart in section III. Subsequently, we delineate two independent schemes
comprising of electrical and bosonic constituents for the realization of the
DSSH model in section IV. In section V, we foreground a protocol for the
realization of DSSH model with non-reciprocal couplings through the
application of a coherent form interaction between bosonic oscillators,
translating into topological transitions and skin effect. Finally, we conclude
our results in section VI.

##  II Key features of the SSH model

We begin by revisiting theSSH model coherently coupled unit cells. To this
end, consider a one-dimensional (1-D) lattice of two different types of sites,
A and B with staggered nearest neighbor couplings as depicted in Fig. 1. The
interaction Hamiltonian of the system subject to open boundary conditions
(OBC) is given by

$\displaystyle
H=\sum_{i=1}^{N}t_{1}\ket{A_{i}}\bra{B_{i}}+\sum_{i=1}^{N-1}t_{2}\ket{A_{{i+1}}}\bra{B_{i}}+h.c,$
(1)

where $N$ denotes the number of unit cells, $\ket{A_{i}}$, $\ket{B_{i}}$
characterize the particle excitation at their respective location while
$t_{1}$ and $t_{2}$ $\in\mathbb{R}$ are the intra and inter-cellular couplings
respectively. Equivalently, we can write the SSH Hamiltonian subject to
periodic boundary conditions (PBC) as

$\displaystyle
H=\sum_{i=1}^{N}t_{1}\ket{A_{i}}\bra{B_{i}}+\sum_{i=1}^{N}t_{2}\ket{A_{f(i)}}\bra{B_{i}}+h.c,$
(2)

where $f(i)=i+1$ mod $N$. Invoking the Bloch theorem, the Hamiltonian under
PBC can be recast in the Fourier domain in terms of the Bloch Hamiltonian
provided by

$\displaystyle H_{k}=\begin{pmatrix}0&R(k)e^{-i\phi(k)}\\\
R(k)e^{i\phi(k)}&0\end{pmatrix},$ (3)

where $k=\frac{2\pi n}{N}$, $m\in\\{1,2..N\\}$,
$R(k)=\sqrt{t_{1}^{2}+t_{2}^{2}+2t_{1}t_{2}\cos{(k)}}$, the phase
$\phi(k)=\arctan(\frac{t_{2}\sin{k}}{t_{1}+t_{2}\cos{k}})$ and we set the
inter-cellular spacing $a=1$. Note that the Hermitian matrix $H_{k}$ is chiral
symmetric, that is $\sigma_{z}{H}_{k}\sigma_{z}=-{H}_{k}$ precipitating in
symmetric eigenvalues $E_{\pm}=\pm R(k)$ and corresponding eigenstates

$\displaystyle\ket{E_{\pm},k}=\frac{1}{\sqrt{2}}\begin{pmatrix}\pm 1\\\
e^{i\phi(k)}\end{pmatrix}.$ (4)

Figure 1: Schematic of the system described by Eqs. (1-2) and Eq. (III) under
open and periodic boundary conditions.

Palpably, the gap between the energy eigenvalues vanishes at $k=\pi$ and
$t_{1}=t_{2}$ as demonstrated in Fig. 2 (a). In contrast, the Hamiltonian
under OBC described by Eq. (1) supports two zero-energy eigenvalues in the
large $N$ limit for $\absolutevalue{\frac{t_{1}}{t_{2}}}<1$, eliciting the
well-known edge modes of the SSH model, a testament to the non-trivial
topology of the system. In addition, one can define a topological invariant,
viz, the winding number $\nu_{\pm}$ defined in terms of the Berry connection
$A_{\pm}(k)=i\bra{E_{\pm},k}\partial_{k}\ket{E_{\pm},k}$ as

$\displaystyle\nu_{\pm}=\frac{1}{\pi}\oint A_{\pm}(k)dk.$ (5)

For instance, $\nu_{+}$ calculated from Eq. (4) and Eq. (5) satisfies

$\displaystyle\nu_{+}=\begin{cases}1&\text{if
$\absolutevalue{\frac{t_{1}}{t_{2}}}<1$}\\\ 0&\text{if
$\absolutevalue{\frac{t_{1}}{t_{2}}}>1$}.\end{cases}$ (6)

Figure 2: (a) The eigenvalues of the Hermitian SSH model described by Eqs.
(1-2) under periodic (green) and open (blue) boundary conditions; (b) The
imaginary part of the eigenvalues of the dissipative SSH model described by
Eq. (III). under periodic (green) and open (blue) boundary conditions and the
number of unit cells $N=25$ and the effective damping
$\Gamma_{r}=\gamma+\Gamma_{1}+\Gamma_{2}$ and $\gamma=3$.

A non-zero winding number $\nu_{\pm}$ is a direct manifestation of the non-
trivial topology of the system demonstrating $2\absolutevalue{\nu_{\pm}}$
number of edge modes and $t_{1}=t_{2}$, the point of vanishing gap between
bulk energy bands demarcates the boundary between the two phases. This is
known as the bulk boundary correspondence (BBC) in Hermitian lattice systems.
It is worth noting that coherent coupling between systems emanates from the
spatial overlap between their respective modes. By contrast, dissipatively
coupled systems with a non-Hermitian form of interaction are prevalent in
nature. In essence, any two systems interacting with a common intermediary
channel will spawn a dissipative form coupling. In the following, we will
discuss the general properties of dissipatively coupled SSH models.

##  III Dissipatively coupled SSH (DSSH) model

Consider a 1-D lattice of sites A and B coupled dissipatively as depicted in
Fig. 1. This is analogous to the Hermitian SSH model, except for a notable
difference in the off-diagonal elements, wherein, the real couplings $t_{1}$
and $t_{2}$ are now replaced by purely imaginary numbers leading to an
effective non-Hermitian Hamiltonian given by

$\displaystyle
H=-\sum_{i=1}^{N}\big{(}(\Delta_{1}-i\Gamma_{r})\ket{A_{i}}\bra{A_{i}}+(\Delta_{2}-i\Gamma_{r})\ket{B_{i}}\bra{B_{i}}\big{)}$
$\displaystyle+\sum_{i=1}^{N}i\Gamma_{1}\ket{A_{i}}\bra{B_{i}}+\sum_{i=1}^{N-1}i\Gamma_{2}\ket{A_{{i+1}}}\bra{B_{i}}+h.c,$
(7)

where $\Gamma_{r}=\gamma+\Gamma_{1}+\Gamma_{2}$ denotes the effective damping
constant corresponding to $A_{i}$ and $B_{i}$ and we assume modes $A_{i}$ and
$B_{i}$ decay at the same rate $\gamma$. The emergence of an effective
Hamiltonian describing the above equation from considerations of a real-space
Hermitian system will be explicated in subsequent sections. For simplicity, we
set $\Delta_{1}=\Delta_{2}=0$. The effective Bloch Hamiltonian under PBC is
provided by

$\displaystyle\mathcal{H}(k)=\begin{pmatrix}-i\Gamma_{r}&i\Gamma_{1}+i\Gamma_{2}e^{-ik}\\\
i\Gamma_{1}+i\Gamma_{2}e^{ik}&-i\Gamma_{r}\end{pmatrix},$ (8)

with eigenvalues $E_{\pm}=-i\Gamma_{r}\pm
i\sqrt{\Gamma_{1}^{2}+\Gamma_{2}^{2}+2\Gamma_{1}\Gamma_{2}\cos{k}}$ where
$k=\frac{2\pi n}{N}$, the parameters $\Gamma_{j}$, $j\in\\{1,2\\}$ are the
absolute value of the strength of dissipative coupling and we have set the
diagonal elements to be identical. Owing to the non-Hermitian nature of the
system, the right eigenvectors and their dual left eigenvector basis of
$\mathcal{H}(k)$ are, in general, not identical. Let $\ket{R_{\pm},k}$ and
$\ket{L_{\pm},k}$ be the bi-orthogonal right and left eigenvectors
respectively of $\mathcal{H}(k)$ defined by

$\displaystyle\mathcal{H}(k)\ket{R_{\pm},k}=E_{\pm}\ket{R_{\pm},k}$
$\displaystyle\mathcal{H}^{\dagger}(k)\ket{L_{\pm},k}=E_{\pm}^{*}\ket{L_{\pm},k},$
(9)

such that $\bra{L_{m},k}\ket{R_{n},k}=\delta_{m,n}$ where $m,n$ $\in\\{+,-\\}$
and $\delta_{m,n}$ is the Kronecker delta. Interestingly, $\mathcal{H}(k)$ is
anti-Hermitian, i.e., $\mathcal{H}^{\dagger}(k)=-\mathcal{H}(k)$. As a result,
the bi-orthogonal eigenvectors have the property

$\displaystyle\ket{R_{\pm},k}=\ket{L_{\pm},k}=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\\
\pm ie^{i\phi(k)},\end{pmatrix}$ (10)

where
$\phi(k)=-\arctan(\frac{\Gamma_{1}+\Gamma_{2}\cos{k}}{\Gamma_{2}\sin{k}})$.
Subsequently, one can define the Berry connection involving the bi-orthogonal
eigenvectors as $A_{\pm}(k)=i\bra{L_{\pm},k}\partial_{k}\ket{R_{\pm},k}$ and
analogous to the Eq. (5) of coherently coupled SSH model, the system is
topological with $\nu_{+}=1$ for $|\Gamma_{2}|>|\Gamma_{1}|$. Note, en
passant, the constant diagonal decay is irrelevant for topological
considerations. An interesting consequence of non-Hermiticity is the breakdown
of BBC. In other words, the parameters corresponding to the energy gap closing
in a non-Hermitian Bloch Hamiltonian do not, in general, signify the boundary
between topological and trivial phases. On the contrary, owing to the anti-
Hermitian nature of $\mathcal{H}(k)$, the dissipatively coupled system
described in Fig. 1 follows BBC. In Fig. 2 (b), we plot the eigenvalues of the
system under PBC and OBC and when
$\absolutevalue{\frac{\Gamma_{1}}{\Gamma_{2}}}<1$, it displays the conspicuous
emergence of two distinct eigenvalues corresponding to the edge modes flanked
on either side by the bulk modes. Not surprisingly, this is exactly the point
of the vanishing energy gap between the Bloch modes corroborating BBC.

Role of real diagonal terms in DSSH: Consider now, the scenario
$\Delta_{1}=-\Delta_{2}=\Delta$. The Bloch Hamiltonian under PBC modifies to

$\displaystyle\mathcal{H}_{k}=\begin{pmatrix}{\Delta}-i\Gamma_{r}&h(k)\\\
-h^{*}(k)&-{\Delta}-i\Gamma_{r}\end{pmatrix},$ (11)

where $h(k)=i\Gamma_{1}+i\Gamma_{2}e^{-ik}$. Observe that the effective
momentum space Hamiltonian is anti-PT symmetric, in other words,
$(\hat{PT})\mathcal{H}_{k}(\hat{PT})=-\mathcal{H}_{k}$. We may rewrite
$h(k)=B_{x}-iB_{y}$, where $B_{x}=\Gamma_{2}\sin(k)$ and
$B_{y}=-(\Gamma_{1}+\Gamma_{2}\cos(k))$ are pseudo magnetic fields providing
an analogy to spin half particles interacting with magnetic fields. When
discussing the topological properties of the system, the constant diagonal
term $-i\Gamma_{r}$ is irrelevant. The eigenvalues of the system, ignoring
$-i\Gamma_{r}$ are given by
$E_{\pm}=\pm\sqrt{\bar{\Delta}^{2}-(B_{x}^{2}+B_{y}^{2})}$ for
$|\bar{\Delta}|>|h(k)|$ and $E_{\pm}=\pm
i\sqrt{B_{x}^{2}+B_{y}^{2}-\bar{\Delta}^{2}}$ for $|\bar{\Delta}|<|h(k)|$.
Here, we focus on the region where $|\bar{\Delta}|<|h(k)|$ and use the
parametrization $\frac{\bar{\Delta}}{R}=\sinh(\theta)$,
$\frac{B_{x}}{R}=\cosh(\theta)\cos(\phi)$,
$\frac{B_{y}}{R}=\cosh(\theta)\sin(\phi)$, where
$R=\sqrt{B_{x}^{2}+B_{y}^{2}-\bar{\Delta}^{2}}$ to obtain the right and left
eigenvectors of the eigenvalue $\lambda_{+}$ as

$\displaystyle\ket{R_{+}}=\frac{1}{\sqrt{2(1+i\sinh(\theta))}}\begin{pmatrix}-i\cosh(\theta)e^{-i\phi}\\\
1+i\sinh(\theta)\end{pmatrix}$ (12)
$\displaystyle\ket{L_{+}}=\frac{1}{\sqrt{2(1+i\sinh(\theta))}}\begin{pmatrix}-i\cosh(\theta)e^{-i\phi}\\\
1-i\sinh(\theta)\end{pmatrix}.$ (13)

The Berry connection of the system is defined as
$A_{+}(k)=i\bra{L_{+}}\partial_{k}\ket{R_{+}}$. Utilizing the parametrization
discussed above, we can recast the Berry connection as
$A(k)=(\partial_{k}\phi)A(\phi)+(\partial_{k}\theta)A(\theta)$. If we
consider, for example, a trajectory in the $\theta$-$\phi$ space, wherein,
$\partial_{k}\theta=0$, the Berry connection is given by
$A(k)=\frac{1}{2}(1-i\sinh(\theta)\partial_{k}\phi$. Note, when $\theta=0$,
the equation for winding number

$\displaystyle\nu_{+}=\frac{1}{\pi}\oint A_{+}(k)dk,$ (14)

morphs into Eq. (5) and the system is topological with $\nu_{+}=1$ for
$|\Gamma_{2}|>|\Gamma_{1}|$. More precisely, the anti-PT symmetric system
demonstrates partial topological phases for trajectories where
$\partial_{k}\theta=0$, wherein, the real part of the winding numbers mimics
the topology of a Hermitian SSH model. It is worth noting that the Hamiltonian
$\mathcal{H}_{k}$ is not chiral symmetric, in other words
$\sigma_{z}\mathcal{H}_{k}\sigma_{z}\neq-\mathcal{H}_{k}$.

In the following section, we provide two experimentally realizable protocols
to engineer DSSH model.

##  IV Realization of the dissipative SSH model

Circuit model: In this section, we provide a circuit model to construct the
DSSH model employing an electrical circuit involving coupled amplifying LRC
resonators connected in parallel through a coupling resistor as depicted in
Fig. 3. Upon solving for Kirchhoff’s equations of motion for voltages, we
obtain

$\displaystyle\ddot{V}_{n}+\omega_{1}^{2}V_{n}+(\gamma_{1}+\Gamma_{1}+\Gamma_{2})V_{n}=\Gamma_{1}\dot{\bar{V}}_{n}+\Gamma_{2}\dot{\bar{V}}_{n-1}$
$\displaystyle\ddot{\bar{V}}_{n}+\omega_{2}^{2}\bar{V}_{n}+(\gamma_{2}+\Gamma_{1}+\Gamma_{2})\bar{V}_{n}=\Gamma_{1}\dot{{V}}_{n}+\Gamma_{2}\dot{{V}}_{n+1}.$
(15)

Figure 3: Circuit model consisting of resistively coupled LCR resonators for
the realization of dissipatively coupled SSH model resulting in the dynamics
described by Eq. (IV).

Here, $\omega_{1}=1/\sqrt{L_{1}C_{1}}$, $\omega_{2}=1/\sqrt{L_{2}C_{2}}$,
$\Gamma_{1}=\frac{1}{R_{c1}C_{1}}$, $\Gamma_{2}=\frac{1}{R_{c2}C_{2}}$,
$\gamma_{i}=\frac{1}{R_{i}C_{i}}$ and $L_{1,2}$, $R_{c}$, $C_{1,2}$ are
respectively inductance, the two coupling resistances, capacitance of the
constituent elements in the unit cell and we assume $C_{1}=C_{2}$. Note en
passant, the constants $\Gamma_{1}$ and $\Gamma_{2}$ represent the intra and
inter-cellular couplings in the lattice. In the weak coupling and small
detunings regime, that is, when
$\\{\Gamma_{1},\Gamma_{2}\\}<<\\{\omega_{1},\omega_{2}\\}$ and
$|\omega_{1}-\omega_{2}|<<\frac{\omega_{1}+\omega_{2}}{2}$, we can reduce the
above equations by using the slowly varying envelope functions
$v_{n}(t),\bar{v}_{n}(t)$, such that

$\displaystyle V_{n}(t)=\frac{v_{n}(t)e^{-i\omega_{0}(t)}+c.c}{2}$
$\displaystyle\bar{V}_{n}(t)=\frac{\bar{v}_{n}(t)e^{-i\omega_{0}(t)}+c.c}{2},$
(16)

where $\omega_{0}=\frac{\omega_{1}+\omega_{2}}{2}$. Employing Eq. (IV), the
dynamics of the envelope functions are obtained as

$\displaystyle\dot{v}_{n}=-i\frac{\bar{\Delta}-i(\gamma_{1}+\Gamma_{1}+\Gamma_{2})}{2}v_{n}+\frac{\Gamma_{1}}{2}\bar{v}_{n}+\frac{\Gamma_{2}}{2}\bar{v}_{n-1}$
$\displaystyle\dot{\bar{v}}_{n}=i\frac{\bar{\Delta}+i(\gamma_{2}+\Gamma_{1}+\Gamma_{2})}{2}\bar{v}_{n}+\frac{\Gamma_{1}}{2}{v}_{n}+\frac{\Gamma_{2}}{2}{v}_{n+1},$
(17)

with $\bar{\Delta}=\frac{\omega_{1}-\omega_{2}}{2}$. We assume propagating
solutions for the sub-lattice elements, that is

$\displaystyle v_{n}=\sum_{k}v_{k,\omega}e^{i(kn-\omega t)}+c.c$
$\displaystyle\bar{v}_{n}=\sum_{k}\bar{v}_{k,\omega}e^{i(kn-\omega t)}+c.c,$
(18)

where $k$ is the wave vector and the lattice constant is taken to be unit.
Substituting Eq. (IV) into Eq. (IV), we arrive at the eigenvalue equation
$(H_{k}-\omega)X=0$, where $X^{T}=$[ $v_{k,\omega}$ $\bar{v}_{k,\omega}$] and

$\displaystyle
H_{k}=\begin{pmatrix}\bar{\Delta}-i(\gamma_{1}+\Gamma_{1}+\Gamma_{2})&i(\frac{\Gamma_{1}}{2}+\frac{\Gamma_{2}}{2}e^{-ik})\\\
i(\frac{\Gamma_{1}}{2}+\frac{\Gamma_{2}}{2}e^{-ik})&-\bar{\Delta}-i(\gamma_{2}+\Gamma_{1}+\Gamma_{2})\end{pmatrix}.$
(19)

The Hamiltonian $H_{k}$ is equivalent to $\mathcal{H}_{k}$. In other words,
the circuit lattice is topologically equivalent to a dissipatively coupled SSH
model.

Photonic systems: We begin by considering the following generic Hamiltonian
comprising of a chain of coherently coupled bosonic sub-lattice elements
$a_{i}$, $b_{i}$, $c_{i}$ and $d_{i}$ under OBC as depicted in Fig. 4.

$\displaystyle\mathcal{H}/\hbar=\sum_{i=1}^{N}\omega_{b,i}b_{i}^{\dagger}b_{i}+\sum_{i=1}^{N}\omega_{c,i}c_{i}^{\dagger}c_{i}+\sum_{i=1}^{N}\omega_{a,i}a_{i}^{\dagger}a_{i}+\sum_{i=0}^{N}\omega_{d,i}d_{i}^{\dagger}d_{i}$
$\displaystyle+\sum_{i=1}^{N}[g_{1}b_{i}^{\dagger}(a_{i}+d_{i-1})+h.c]+\sum_{i=1}^{N}[g_{2}c_{i}^{\dagger}(a_{i}+d_{i})+h.c]$
(20)

Here, $\omega_{x,i}$ characterizes the resonance frequencies of the modes
$x_{i}$, where $x\in\\{a,b,c,d\\}$ and $g_{1}$, $g_{2}$ $\in\mathbb{R}$ are
the strength of dispersive coupling between the modes.

Figure 4: Schematic of the coupled oscillator system described by the
Hamiltonian in Eq. (IV) comprising of a bath of oscillators coupled with a
system of otherwise non-interacting bosonic modes.

We assume that all the modes $b_{i}$ and $c_{i}$ decay at approximately the
same rate $\gamma$ whereas the modes $a_{i}$ and $d_{i}$ decay at rates
$\kappa_{1}$ and $\kappa_{2}$ respectively. Further, we set
$\Delta_{b,i}=\Delta_{1}=\omega_{b,1}-\omega_{a,1}$,
$\Delta_{c,i}=\Delta_{2}=\omega_{c,1}-\omega_{a,1}$,
$\omega_{a,i}=\omega_{b,i}=\omega_{a,1}$ and $\kappa_{1}>\kappa_{2}$. In the
weak coupling domain, that is, when the leakage rates $\kappa_{i}$ strongly
dominates the dynamics of the system, in other words,
$\\{g_{1},g_{2},\gamma,\Delta_{b,1},\Delta_{c,1}\\}<<\\{\kappa_{1},\kappa_{2}\\}$,
we can adiabatically eliminate the $a_{i}$ and $d_{i}$ modes resulting in the
effective momentum space Hamiltonian of the system under PBC ($d_{0}=d_{N}$)
in the frame rotating at $(\Delta_{1}+\Delta_{2})/2$ as

$\displaystyle\mathcal{H}_{k}=\begin{pmatrix}\bar{\Delta}-i\Gamma_{r}&h(k)\\\
-h^{*}(k)&-\bar{\Delta}-i\Gamma_{r}\end{pmatrix},$ (21)

where, we set $g_{1}=-g_{2}=g$, $\bar{\Delta}=(\Delta_{1}-\Delta_{2})/2$,
$\Gamma_{r}=\gamma+\Gamma_{1}+\Gamma_{2}$,
$h(k)=i\Gamma_{1}+i\Gamma_{2}e^{-ik}$ and $k$ is the lattice constant
equivalent to Eq. (11) and $\Gamma_{i}=\frac{g^{2}}{\kappa_{i}}$. Note that
the pair of modes $b_{i}$ and $c_{i}$ form a unit cell with intra and inter
cell couplings $i\frac{g^{2}}{\kappa_{1}}$ and $i\frac{g^{2}}{\kappa_{2}}$
respectively equivalent to the system in Fig. 1 under dissipative settings
with $\Gamma_{i}$ replaced by $\frac{g^{2}}{\kappa_{i}}$.

##  V Non-reciprocity, phase-dependent topological transitions and skin effect

In the previous section, we briefly mentioned the breakdown of BBC in non-
Hermitian systems. In particular, non-reciprocal (chiral) coupling between
sub-lattice elements under OBC culminates in the skin effect which is the
exponential localization of right and left eigenvectors at the lattice
boundaries without any distinction between bulk and edge modes. In addition,
the points in the parametric space corresponding to the closing of energy gap
under PBC do not indicate the emergence of eigenmodes with eigenvalues
$-i\Gamma_{r}$ under OBC requiring a bi-orthogonal modification of the BBC. In
the following, we discuss the construction of non-reciprocal couplings in DSSH
lattice leading to phase-dependent topological transitions and skin effect.

Consider now, a one-dimensional lattice of bosonic modes $b$, $c$ coupled with
auxiliary modes $a$ as depicted in Fig. 4 where we have now switched on the
coherent coupling between $b_{i}$ and $c_{i}$ modes. The system is
characterized by the Hamiltonian

$\displaystyle\tilde{\mathcal{H}}/\hbar=\mathcal{H}/\hbar+\sum_{i=1}^{N}[Gb_{i}^{\dagger}c_{i}+h.c],$
(22)

where $\mathcal{H}$ is given by Eq. (IV) and $G=|G|e^{i\alpha}$. Once again,
in the weak coupling domain, that is, when $\\{g_{1},g_{2},|G|,\gamma\\}$ are
significantly less than the cavity leakage $\kappa_{i}$ and setting
$\Delta_{b,i}=\Delta_{1}=\omega_{b,1}-\omega_{a,1}$,
$\Delta_{c,i}=\Delta_{2}=\omega_{c,1}-\omega_{a,1}$,
$\omega_{a,i}=\omega_{b,i}=\omega_{a,1}$, $\kappa_{1}>\kappa_{2}$ and
$g_{1}=-g_{2}=g$, we can obtain an effective system between modes $b_{i}$ and
$c_{i}$. The system under PBC translates into the following Bloch Hamiltonian
in the frame rotating at $(\Delta_{1}+\Delta_{2})/2$

$\displaystyle\mathcal{H}_{k}=\begin{pmatrix}\bar{\Delta}-i\Gamma_{r}&i\Gamma_{-}+i\Gamma_{2}e^{-ik}\\\
i\Gamma_{+}+i\Gamma_{2}e^{ik}&-\bar{\Delta}-i\Gamma_{r}\end{pmatrix},$ (23)

where $\Gamma_{\pm}=\Gamma_{1}\mp|G|\sin{\alpha}-i|G|\cos{\alpha}$,
$\bar{\Delta}=(\Delta_{1}-\Delta_{2})/2$ and
$\Gamma_{r}=\gamma+\Gamma_{1}+\Gamma_{2}$. Notice, the conspicuous emergence
of a purely dissipative form of non-reciprocal coupling between the subsystems
when $\alpha=\pi/2$.

Note that the non-Hermitian system described by the aforementioned Hamiltonian
does not follow BBC. To elucidate this in detail, let us begin by considering
the system under OBC. Before expounding the analysis of the full system, it is
worthwhile to explicate the properties of the system in the absence of $c_{N}$
and to simplify the analysis, we set $\bar{\Delta}=0$ for the remaining part
of this section. When the lattice terminates in $b_{N}$, the $2N-1$
dimensional Hamiltonian of the system $H_{broken}^{b}$ supports bi-orthogonal
eigenstates with eigenvalue $-i\Gamma_{r}$ of the form [32, 61]

$\displaystyle\ket{R}^{b}={N}_{R}^{b}\sum_{n=0}^{N-1}\Big{(}-\frac{\Gamma_{2}}{\Gamma_{+}}\Big{)}^{N-n}b_{n+1}^{\dagger}\ket{0}$
$\displaystyle\ket{L}^{b}={N}_{L}^{b}\sum_{n=0}^{N-1}\Big{(}-\frac{\Gamma_{2}}{\Gamma_{-}^{*}}\Big{)}^{N-n}b_{n+1}^{\dagger}\ket{0},$
(24)

where $N_{R}$, $N_{L}$ are normalization constants such that
$\bra{L}\ket{R}=1$ provided by
$N_{L}^{b*}N_{R}^{b}=Z^{N+1}\frac{(Z^{-1}-1)}{1-Z^{N}}$,
$Z=\frac{\Gamma_{-}\Gamma_{+}}{\Gamma_{2}^{2}}$ and

$\displaystyle
H_{broken}^{b}=\begin{bmatrix}-i\Gamma_{r}&i\Gamma_{-}&0&0&\dots&0\\\
i\Gamma_{+}&-i\Gamma_{r}&i\Gamma_{2}&0&\dots&0\\\
0&i\Gamma_{2}&-i\Gamma_{r}&i\Gamma_{-}&\dots&0\\\
0&0&i\Gamma_{+}&-i\Gamma_{r}&\dots&0\\\ 0&0&\dots&\dots&\dots&i\Gamma_{2}\\\
0&0&\dots&\dots&i\Gamma_{2}&-i\Gamma_{r}\end{bmatrix}_{2N-1}.$ (25)

This is due to destructive interference at $c_{i}$ sites and observe that
$\ket{R}^{b}$ and $\ket{L}^{b}$ can be written as a column matrix, for
instance,

$\displaystyle\ket{R}^{b}=\begin{bmatrix}\Big{(}-\frac{\Gamma_{2}}{\Gamma_{+}}\Big{)}^{N}&\\\
0\\\ \Big{(}-\frac{\Gamma_{2}}{\Gamma_{+}}\Big{)}^{N-1}\\\ \vdots\\\ 0\\\
-\frac{\Gamma_{2}}{\Gamma_{+}}\end{bmatrix}_{2N-1},$ (26)

and clearly $H_{broken}^{b}\ket{R}^{b}=-i\Gamma_{r}\ket{R}^{b}$. Notice that
when $\alpha=\pi/2$, Eq. (V) morphs into

$\displaystyle\ket{R}^{b}={N}_{R}^{b}\sum_{n=0}^{N-1}\Big{(}-\frac{\Gamma_{2}}{\Gamma_{1}+\absolutevalue{G}}\Big{)}^{N-n}b_{n+1}^{\dagger}\ket{0}$
$\displaystyle\ket{L}^{b}={N}_{L}^{b}\sum_{n=0}^{N-1}\Big{(}-\frac{\Gamma_{2}}{\Gamma_{1}-\absolutevalue{G}}\Big{)}^{N-n}b_{n+1}^{\dagger}\ket{0},$
(27)

which is, identical to the results for a coherently coupled SSH model with
non-reciprocal intra-cell couplings. By the same token, one can construct
eigenstates of complex energy $-i\Gamma_{r}$ of the Hamiltonian
$H_{broken}^{c}$ of the lattice which is broken at the other end, i.e., when
the lattice ends on either side with $c_{i}$ sites as

$\displaystyle\ket{R}^{c}={N}_{R}^{c}\sum_{n=1}^{N}\Big{(}-\frac{\Gamma_{2}}{\Gamma_{-}}\Big{)}^{N}c_{n}^{\dagger}\ket{0}$
$\displaystyle\ket{L}^{c}={N}_{L}^{c}\sum_{n=1}^{N}\Big{(}-\frac{\Gamma_{2}}{\Gamma_{+}^{*}}\Big{)}^{N}c_{n}^{\dagger}\ket{0},$
(28)

where
$N_{L}^{c*}N_{R}^{c}=N_{L}^{b*}N_{R}^{b}=Z^{N+1}\frac{(Z^{-1}-1)}{1-Z^{N}}$.

In stark contrast to Hermitian systems where the absolute value square of the
coeffcient of the column matrix in Eq. (26) would represent the probability of
finding the excitation at the $n^{th}$ unit cell, non-Hermitian systems
necessitate a bi-orthogonally defined projection to the $n^{th}$ unit cell.
Subsequently, one can define a biorthogonal projection operator $P_{n}$ to the
$n^{th}$ unit cell of the lattice as
$P_{n}=\ket{b,n}\bra{b,n}+\ket{c,n}\bra{c,n}$ where
$\ket{b,n}=b_{n}^{\dagger}\ket{0}$ and $\ket{c,n}=c_{n}^{\dagger}\ket{0}$. For
example, projecting the states in Eq. (V) on to the $n^{th}$ unit cell
provides

$\displaystyle\bra{L}^{b}P_{n}\ket{R}^{b}=\frac{Z^{n+1}(Z^{-1}-1)}{1-Z^{N}}.$
(29)

It is, therefore, apparent that for $|Z|<1$, the excitation is exponentially
localized at the left edge ($n=1$) whereas $|Z|>1$ localizes the state at the
right edge $n=N$. Similarly, when the lattice terminates with a $c_{i}$ mode
on either side ($b_{1}$ is absent), one can obtain analogous results with the
excitation localized at the right (left) edge for $|Z|<1(|Z|>1)$ due to mirror
symmetry.

The results of the broken chain system can now be used to extract the physics
of the full system in the thermodynamic limit (large $N$). Consider the
Hamiltonian of the full system

Figure 5: The absolute value of eigenvalues of $H_{full}$ under OBC (blue) and
PBC (green) for different values of $\alpha$ and $|G|$ with $\Gamma_{2}=2$,
$\gamma=3$, the number of units cells $N=25$. The vertical lines represent the
points $x_{\pm}$ obtained from Eq. (34) as $x_{\pm}=\sqrt{A_{\pm}}$ for non-
negative values of $A_{\pm}$. The two states with complex energy
$-i\Gamma_{r}=-i(\gamma+\Gamma_{1}+\Gamma_{2})$ appear in the region
$\Gamma_{1}<x_{-}$ and $\Gamma_{1}<x_{+}$ as isolated blue lines in the middle
in (a), (b), whereas they appear in the region $x_{-}<\Gamma_{1}<x_{+}$ in
(d).

$\displaystyle
H_{full}=\begin{bmatrix}-i\Gamma_{r}&i\Gamma_{-}&0&0&\dots&0&0\\\
i\Gamma_{+}&-i\Gamma_{r}&i\Gamma_{2}&0&\dots&0&0\\\
0&i\Gamma_{2}&-i\Gamma_{r}&i\Gamma_{-}&\dots&0&0\\\
0&0&i\Gamma_{+}&-i\Gamma_{r}&\dots&0&0\\\
0&0&\dots&\dots&\dots&i\Gamma_{2}&0\\\
0&0&\dots&\dots&i\Gamma_{2}&-i\Gamma_{r}&i\Gamma_{-}\\\
0&0&\dots&\dots&0&i\Gamma_{+}&-i\Gamma_{r}\end{bmatrix}_{2N},$ (30)

and states

$\displaystyle\ket{\psi}_{R}^{\pm}=\frac{1}{\sqrt{2}}\Biggl{\\{}\begin{bmatrix}\ket{R}^{b}\\\
0\end{bmatrix}\pm\begin{bmatrix}0\\\ \ket{R}^{c}\end{bmatrix}\Biggl{\\}}$
$\displaystyle\ket{\psi}_{L}^{\pm}=\frac{1}{\sqrt{2}}\Biggl{\\{}\begin{bmatrix}\ket{L}^{b}\\\
0\end{bmatrix}\pm\begin{bmatrix}0\\\ \ket{L}^{c}\end{bmatrix}\Biggl{\\}}.$
(31)

Figure 6: The phase diagram of the system as a function of $\alpha$ and $|G|$
for $\Gamma_{2}=2$ characterized by the real positive values of $A_{\pm}$
demarcating the topological boundaries. In particular, the region in red
depicts the topologically trivial parametric domain.

Clearly, $\bra{\psi}_{L}^{\pm}\ket{\psi}_{R}^{\pm}=1$ and
$\bra{\psi}_{L}^{\mp}\ket{\psi}_{R}^{\pm}=0$ and it is straightforward to
obtain

$\displaystyle H_{full}\ket{\psi}_{R}^{\pm}=\begin{bmatrix}\mp
i\Gamma_{2}N_{R}^{c}&\\\ 0\\\ \vdots\\\ 0\\\
-i\Gamma_{2}N_{R}^{b}\end{bmatrix}_{2N}-i\Gamma_{r}\ket{\psi}_{R}^{\pm}$
$\displaystyle H_{full}^{\dagger}\ket{\psi}_{L}^{\pm}=\begin{bmatrix}\pm
i\Gamma_{2}N_{L}^{c}&\\\ 0\\\ \vdots\\\ 0\\\
i\Gamma_{2}N_{L}^{b}\end{bmatrix}_{2N}+i\Gamma_{r}\ket{\psi}_{L}^{\pm},$ (32)

where we have defined
$N_{R}^{b}=N_{R}^{c}=\Big{(}\frac{Z^{N+1}(Z^{-1}-1)}{1-Z^{N}}\Big{)}^{1/2}$
and $N_{L}^{b}=N_{L}^{c}=N_{R}^{b*}$. It is conspicuous from the above
expression that $\ket{\psi}_{R,L}^{\pm}$ represent bi-orthogonal eigenstates
of $H_{full}$ with complex energy $-i\Gamma_{r}$ for large $N$ for
$N_{R}^{b}\rightarrow 0$, viz, if $|Z|<1$ (and not for $|Z|\geq 1$) as the
normalization factors in the first part of the RHS of Eq. (V) approach zero.
In other words, the states in Eq. (V) are the eigenstates of $H_{full}$ with
eigenvalue $-i\Gamma_{r}$ ($\Gamma_{r}$-modes) for

$\displaystyle\sqrt{(\Gamma_{1}^{2}-\absolutevalue{G}^{2})^{2}+4\absolutevalue{G}^{2}\Gamma_{1}^{2}\cos^{2}{\alpha}}<\Gamma_{2}^{2}.$
(33)

It is worth noting that for $\alpha=\pi/2$, the condition for bi-orthogonal
edge modes modifies to $\Gamma_{1}^{2}-\absolutevalue{G}^{2}<\Gamma_{2}^{2}$.
The Eq. (33) may be rewritten as
$(\Gamma_{1}^{2}-A_{+})(\Gamma_{1}^{2}-A_{-})<0$, where

$\displaystyle
A_{\pm}=-|G|^{2}\cos{(2\alpha)}\pm\sqrt{\Gamma_{2}^{4}-|G|^{4}\sin^{2}{(2\alpha)}}$
(34)

for real values of the RHS of Eq. (34). Therefore, the $\Gamma_{r}$-modes of
the system under OBC occur in the region where $\Gamma_{1}^{2}<A_{+}$ and
$\Gamma_{1}^{2}>A_{-}$. It is worth noting that the system does not incur
$\Gamma_{r}$-modes for $\Gamma_{2}^{4}-|G|^{4}\sin^{2}{(2\alpha)}<0$. In Fig.
5, we plot the absolute value of eigenvalues of the full system under OBC and
PBC for different values of $\alpha$ and $|G|$ for $\Gamma_{2}=2$ where we
have defined $x_{\pm}=\sqrt{A_{\pm}}$ for non-negative values of $A_{\pm}$. In
Fig. 5 (a-b), for $|G|=1$, we observe that $A_{+}>0$ and $A_{-}<0$ lending
$\Gamma_{r}$-modes for $|\Gamma_{1}|<x_{+}$ depicted by the two isolated blue
lines. In stark contrast, for $|G|=3$ and $\alpha=\pi/4$,
$\Gamma_{2}^{4}-|G|^{4}\sin^{2}{(2\alpha)}<0$, leading to the conspicuous
absence of $\Gamma_{r}$-modes as depicted in Fig. 5 (c). However,
$\alpha=\pi/2$, $|G|=3$ (Fig. 5 (d)) renders $A_{\pm}>0$ which affords
$\Gamma_{r}$-modes on either in the region $x_{-}<\Gamma_{1}<x_{+}$,
demonstrating phase dependent nature of topological transitions. Note that the
green curves in Fig. 5 correspond to the absolute value of energy under OBC.
Clearly, the points where the blue curves approach $-i\Gamma_{r}$ do not match
with that of the green curves as a consequence of the breakdown of BBC. In
Fig. 6, we plot the phase (not to be confused with $\alpha$) diagram of the
system as a function of $\alpha$ and $|G|$ depending on the real positive
values of $A_{\pm}$, clearly demarcating the topological boundaries. For the
region depicted in yellow, we have only the $A_{+}\geq 0$, begetting
$\Gamma_{r}$-modes modes for $|\Gamma_{1}|<\sqrt{A_{+}}$ as displayed in Fig.
5 (a-b). The region in blue, however, provides $A_{\pm}\geq 0$, lending
$\Gamma_{r}$-modes when $\Gamma_{1}^{2}<A_{+}$ and $\Gamma_{1}^{2}>A_{-}$ as
demonstrated in Fig. 5 (d). In contrast, the region in red prohibits real
positive values for $A_{\pm}$ and therefore does not lend itself to a
topological description evident from Fig. 5 (c).

To provide more substance to the above discussion, we plot, in Fig. 7 (a-b)
the absolute value of $N$ components (equally spaced between 0 and 1) of the
vector $V^{i}=$[ $\pi_{1}^{i}$ $\pi_{2}^{i}$ … $\pi_{N}^{i}$ ] for two
different regions of Fig. 5(d) with $i\in\\{1,2,..N\\}$ and

Figure 7: (a) The absolute value of $N$ components (equally spaced between 0
and 1) of the vector $V^{i}$ for $i=1$ to $5$; (b) absolute value of $N$
components of $V^{i}$ for $i=1$ to $N$. Note that the red curves in (b)
represent the two topological edge-modes of the system (concentrated at both
the right and left edges) and we have set the number of unit cells $N=25$,
$\Gamma_{2}=2$, $\gamma=3$. All the $N$ vectors $V^{i}$ are bulk-modes for
$\alpha=\pi/2$, $|G|=3$ and $\Gamma_{1}=0.5$ and for clarity, we only plot
five of them in (a) as the rest of them have similar behavior.

$\displaystyle\pi_{n}^{i}=\frac{\bra{L_{i}^{full}}P_{n}\ket{R_{i}^{full}}}{\bra{L_{i}^{full}}\ket{R_{i}^{full}}}$
(35)

Figure 8:  (a) The absolute value of the eigenvectors (equally spaced between
0 and 1) $\ket{R_{i}^{full}}$ for $\alpha=\pi/2$, $|G|=3$, $\Gamma_{1}=0.5$,
clearly demonstrating the eigenmodes, all of which are concentrated on the
left edge of the system; (b) absolute value of the eigenvectors
$\ket{R_{i}^{full}}$ for $\alpha=0$, $|G|=3$, $\Gamma_{1}=0.5$ with no non-
reciprocity in couplings leading to the absence of edge effects; (c) plot of
the absolute value of the eigenvectors $\ket{R_{i}^{full}}$ (for the region
between zero and 0.15) with $\alpha=\pi/2$, $|G|=3$, $\Gamma_{1}=3$ with
$\Gamma_{+}\rightarrow 0$ giving rise to extreme non-reciprocity and
amplification of edge population when compared to (a). The quantity
$\absolutevalue{R_{i}^{full}}$ is negligibly small beyond $0.15$ in (c) and
analogous to Fig. 7 (a), for clarity, we have only plotted $V^{i}$ for $i=1$
to $5$ in (b). The number of unit cells $N=25$, $\Gamma_{2}=2$, $\gamma=3$.

where $\ket{R_{i}^{full}}$, $\ket{L_{i}^{full}}$ are the right and
eigenvectors of $H_{full}$ respectively. The Fig. 7 (a) corresponds to a point
in the topologically trivial region $A$ of the Fig. 5(d), that is, for
$\alpha=\pi/2$, $\Gamma_{1}=0.5$, $|G|=3$ hallmarked by the absence of edge
modes. In contrast, the Fig. 7 (b) depicts the absolute value of the
projection for $\alpha=\pi/2$, $\Gamma_{1}=2.7$, $|G|=3$, a point in the
region $B$ of Fig. 5(d). Not surprisingly, edge states (in red) springs into
existence and they correspond to Eq. (V) matching the two isolated central
modes of Fig. 5(d) possessing complex energy $-i\Gamma_{r}$. To be precise,
the two edge-modes in Fig. 7 (b) correspond to $V^{i}$ where
$\ket{R_{i}^{full}}$, $\ket{L_{i}^{full}}$ are replaced, respectively, by the
states $\ket{\psi}_{R}^{\pm}$ and $\ket{\psi}_{L}^{\pm}$. On the contrary, the
absolute value of the components of eigenvectors $\ket{R_{i}^{full}}$ and
$\ket{L_{i}^{full}}$ would provide starkly dissimilar results with all the
eigenstates concentrated at the boundaries, otherwise known as the skin
effect. Skin effect, viz, the unusual localization of a large number of
eigenstates at the boundaries [29], a consequence of non-reciprocal coupling,
leads to pronounced sensitivity of the bulk to boundary conditions. In Fig.
8(a), we plot $\absolutevalue{R_{i}^{full}}$, the absolute value of the $N$
components of the column matrix (equally spaced between 0 and 1) representing
$\ket{R_{i}^{full}}$ for $\alpha=\pi/2$ clearly demonstrating the accumulation
of eigenstates at the boundaries of the system owing to the non-reciprocal
nature of dissipative coupling. On the contrary, $\alpha=0$ does not incur any
non-reciprocity in coupling, culminating in the absence of skin effect as
depicted in Fig. 8 (b). In stark contrast to Fig. 8 (a, b), the condition
$\alpha=\pi/2$ and $|G|=\Gamma_{1}$ results in $\Gamma_{+}\rightarrow 0$,
i.e., extreme non-reciprocity and skin-effect. This is manifested in Fig. 8
(c), showcasing the remarkably high localization of the right eigenvectors at
the left edge.

##  VI conclusions

In conclusion, we considered SSH models with a dissipative form of coupling
between the subsystems and discussed some of the interesting physics ensuing
from such models. In particular, we provided two distinct schemes for the
realization of DSSH models in the context of bosonic systems and electrical
LCR resonators. We showed that a collection of resistively coupled LCR
resonators mimic the topology of DSSH models by solving the Kirchhoff’s
equation for voltages. In the framework of bosonic systems, we observed that a
system of non-interacting oscillators interacting with an engineered bath of
modes possessing considerably small lifetimes compared to other system
parameters is equivalent to a DSSH model. Further, by enabling the coherent
interaction between the oscillators under consideration, we showed that the
system affords non-reciprocal dissipative couplings eliciting topological
transitions governed by the phase of the coherent interaction strength and
skin effect. Note that our analyses are generic, relevant to a large class of
systems, especially in microwave to optical settings and merits immediate
realization in the experiments.

##  VII Acknowledgements

GSA and MOS acknowledge the support of the Air Force Office of Scientific
Research [AFOSR award no FA9550-20-1-0366 DEF] and the Robert A. Welch
Foundation (Grant No. A-1261, Grant no A-1943). JMPN acknowledges the support
of the Herman F. Heep and Minnie Belle Heep Texas A&M University endowed fund
held/administered by the Texas A&M Foundation.

##  Appendix A Kirchhof’s equations for the circuit DSSH model

We begin by considering the two blocks of LCR circuits, in other words, a
dimer, coupled through a resistor as depicted in Fig. 9. After the choice of
direction of currents as illustrated in the figure, we use the well-known
Kirchoff’s circuit laws to explicate the dimer dynamics.

Figure 9: The unit cell of the circuit SSH model described in Fig. 3. The blue
arrows represent the chosen direction currents in the system.

$\displaystyle
I_{R_{c1}}=I_{L_{1}}+I_{C_{1}}+I_{R_{1}}=-(I_{L_{2}}+I_{C_{2}}+I_{R_{2}}),$
(36) $\displaystyle
V_{n}=-L_{n}\frac{dI_{L_{n}}}{dt}=-\frac{1}{C_{n}}\int_{0}^{t}I_{C_{n}}(t^{\prime})dt^{\prime}=-I_{R_{1}}R_{1},$
(37) $\displaystyle V_{1}-V_{2}=I_{R_{c1}}R_{c1},$ (38)

where n=1,2. Employing Eq. (37) and Eq. (38) into Eq. (36), we obtain

$\displaystyle\ddot{V_{i}}+\frac{1}{C_{i}}\Big{(}\frac{1}{R_{c1}}+\frac{1}{R_{i}}\Big{)}\dot{V_{i}}+\frac{V_{i}}{L_{i}C_{i}}=\frac{\dot{V_{j}}}{R_{c1}C_{i}},\quad
i\neq j.$ (39)

Upon redefining $\omega_{i}=\frac{1}{\sqrt{L_{i}C_{i}}}$,
$\Gamma_{i}=\frac{1}{R_{c1}C_{i}}$ and $\gamma_{i}=\frac{1}{R_{i}C_{i}}$ Eq.
(39) reduces to

$\displaystyle\ddot{V}_{i}+(\gamma_{i}+\Gamma_{i})V_{i}+\omega_{i}^{2}V_{i}=\Gamma_{i}\dot{V_{j}},\quad
i\neq j,$ (40)

The Eq. (40) can be further simplified if we assume
$V_{i}(t)=\frac{1}{2}u_{i}(t)e^{-i\omega_{0}t}+c.c$, where
$\omega_{0}=\frac{1}{2}(\omega_{1}+\omega_{2})$ and $u_{i}(t)$ is a slowly
varying envelope. In addition, we assume that $k<<\omega_{i}$, $\omega_{1}$
close to $\omega_{2}$ and $C_{1}=C_{2}$. Under these conditions, the Eq. (40)
can be approximated to

$\displaystyle\begin{pmatrix}\dot{u_{1}}\\\
\dot{u_{2}}\end{pmatrix}=-i\frac{1}{2}\begin{pmatrix}\omega_{1}-\omega_{2}-i(\gamma_{1}+\Gamma_{1})&i\Gamma_{1}\\\
i\Gamma_{1}&\omega_{2}-\omega_{1}-i(\gamma_{2}+\Gamma_{1})\end{pmatrix}\begin{pmatrix}{u_{1}}\\\
{u_{2}}\end{pmatrix}.$ (41)

Extending the analysis to the full system in Fig. 3, we obtain Eq. (IV).

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