# Topological Classification of One-Dimensional Chiral Symmetric Interfaces

Harry MullineauxSanders  SUPA, School of Physics and Astronomy, University of
St. Andrews, North Haugh, St. Andrews KY16 9SS, United Kingdom  Bernd
Braunecker  SUPA, School of Physics and Astronomy, University of St. Andrews,
North Haugh, St. Andrews KY16 9SS, United Kingdom

###### Abstract

We address the topological classification of one-dimensional chiral symmetric
interfaces embedded into a two-dimensional substrate. A proof of the validity
of a topological classification based on the Green’s function by explicit
evaluation of the topological invariant is presented. Further, we show that
due to entanglement between the in-gap modes and the substrate, the full
physics of the substrate that is contained in the Green’s function is
required. This is done by considering a classification scheme derived from the
reduced ground state projector, for which we show that an uncritical handling
produces erroneous changes in the topological index due to entanglement driven
gap closures. We illustrate our results by applying them to a tight-binding
model of a spiral magnetic interface in a s-wave superconductor.

The realisation of topological quantum states has been a major aim of
condensed matter physics for a number of years. Most prominent is the task of
obtaining robust Majorana modes at the edge of materials or in vortex cores
[1, 2, 3], as they also would provide the first steps towards topological
quantum computation [4, 5]. In the absence of topological properties in a bulk
material [6], artificially created topological superconductors in
superconductor-semiconductor structures [7] or magnetic-superconductor
interfaces [8] have emerged as promising candidates for the creation of
tunable topological states. The latter, specifically chains of magnetic
impurities on standard s-wave superconductors, have been a source of great
interest in recent years.

Figure 1: Sketch of a generic spiral magnetic interface model: A square
lattice tight-binding model of a s-wave superconducting substrate with a chain
of spiralling magnetic impurities along the $x$ direction.

Magnetic impurities locally break apart Cooper pairs, lowering the
superconducting gap [9, 10] and creating bound Yu-Shiba-Rusinov electronic
states [11, 12, 13]. Chains of impurities, such as shown in Fig. 1, then
create in-gap bands [12] which can be taken through a topological phase
transition by tuning the magnetic scattering strength and the magnetic
alignment. The topological properties of these kinds of magnetic interfaces
have been determined by exact diagonalisation of tight-binding models [8, 9,
14, 15, 16], by first principles numerical calculation of the Green’s function
[17, 18, 19], by analytic determination of the Green’s function with a Debye
frequency cut off [20], by effective tight-binding hybridisation of the
impurity states, [21, 22, 23, 24, 25, 26], and by modelling a one-dimensional
(1D) magnetic wire placed in proximity to a superconductor which can be
integrated out [27, 28]. In the case of densely packed impurity chains, the
effect of the dimensionally embedded nature of the interface on the band
structure and topological properties has been derived from an exact
determination of the Green’s function [29, 30, 31, 32, 33]. There has also
been significant experimental effort in fabricating and studying these
interfaces, some of which have shown signs of topological physics [34, 35, 36,
37, 38, 39, 40, 41].

As the topological properties are due to the 1D interface, in many approaches
topological invariants are derived from an effective 1D Hamiltonian. However
it was shown in Ref. [31], that any topological classification based on
integrating the transverse spatial degrees of freedom out of the Hamiltonian
is only valid in the specific limit of tightly confined impurity bound modes,
which typically is not realised in experiment [36]. The central problem with
such a tracing over the transverse spatial dimension is that essential
information is lost or falsified if the in-gap mode has entanglement between
its inter-unit-cell and spatial degrees of freedom. This is at the risk that
topological properties are not correctly reproduced [31]. A further
complication from the large spatial extend of the in-gap states is that a
simple modelling based on the hybridisation of single impurity modes is
insufficient to capture essential features such as gap closures around the
Fermi momentum or dimensional renormalisation of the scattering strength at
phase transitions [30]. In the absence of an effective 1D Hamiltonian from
which an effective ground state projector can be derived, standard topological
classification methods are unavailable. Therefore topological properties must
be derived from an (ideally analytically and experimentally) available
quantity. Such classification schemes have been proposed but only with
circumstantial conjecture of their validity.

In this paper, we provide a general proof of validity of an appropach based on
local Green’s functions [31, 32], and extend its validity to any chiral
symmetry class. To illustrate the scheme’s applicability we use it to obtain
the full topological phase diagram of a tight binding model of a magnetic
interface embedded in a superconductor, by exactly solving for the Green’s
function. To demonstrate further that this method is resilient against the
aforementioned reduction of spatial degrees of freedom, we compare it with the
reduced ground state projector approach of Ref. [42]. We demonstrate that in
the parameter regions of most interest, the two-dimensional (2D) extension of
the in-gap modes is significant enough that the latter approach requires
careful handling to avoid artefacts of the truncation.

Topological Hamiltonian – We consider topological invariants based on an
effective topological Hamiltonian obtained form the local Green’s function.
This has been proposed before [31, 32] and observed to be consistent with the
correct topological classification but without the general proof of validity
which we shall provide.

To this end we consider a gapped 2D translationally invariant substrate with
Hamiltonian $\mathcal{H}_{0}(k_{x},k_{y})$, for momenta $k_{x},k_{y}$, and
energies chosen such that the gap is centred about energy $\omega=0$. A
scattering interface is introduced as a strip along the $x$ direction that is
bounded in the $y$ direction. For simplicity, we take the interface to be only
at $y=0$ (such as in Fig. 1), however our results generalise straightforwardly
to wider interfaces. In such a case case, the $y$ dimension is partitioned
into segments of the width of the interface, and the $y$ coordinate then
labels each segment whereas the spatial dependence within a segment is taken
as an internal degree of freedom. Assuming translational invariance along the
$x$ direction, we define the projector onto the interface as $P_{y=0}$ and its
scattering Hamiltonian $\mathcal{H}_{V}(k_{x})=H_{V}(k_{x})\otimes P_{y=0}$
where $H_{V}$ acts on the remaining degrees of freedom.

We denote by $g(\omega,k_{x},y)$ the retarded Green’s function at energy
$\omega$ of $\mathcal{H}_{0}(k_{x},k_{y})$ after the partial Fourier transform
$k_{y}\rightarrow y$. Then the total retarded Green’s function of
$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{V}$ with broken translational
invariance in the $y$ direction is given by the Dyson equation [43]

$G(\omega,k_{x},y,y^{\prime})=\\\
g(\omega,k_{x},y-y^{\prime})+g(\omega,k_{x},y)T(\omega,k_{x})g(\omega,k_{x},-y^{\prime}),$
(1)

where the effect of $H_{V}$ is contained in the $T$-matrix

$\displaystyle T(\omega,k_{x})=[H_{V}^{-1}-g(\omega,k_{x},0)]^{-1}.$ (2)

If bound states appear at the interface, their energies are given by the poles
of $T$ as a function of $\omega$, and the corresponding residues provide
information about their wavefunctions. Consequently $T$, and thus $G$, are
expected to contain all topological information too, and it was proposed in
Refs. [31, 32] that

$\displaystyle H^{top}(k_{x},y)=-[G(\omega=0,k_{x},y,y)]^{-1}$ (3)

forms a suitable topological Hamiltonian for the topological classification.
Similar Hamiltonians have been used to study interacting topological materials
[44, 45]. When $y=0$, the chiral invariant of this topological Hamiltonian for
a spiral magnetic superconductor interface has been shown to match the
Pfaffian invariant calculated from a tight-binding model [32] and the presence
of Majorana edge modes [31]. For $y\neq 0$, zeros of the Green’s function
create fictitious phase boundaries which perfectly cancel the true phase
boundaries as $|y|\rightarrow\infty$ so $H^{top}(|y|\rightarrow\infty)$ is
always trivial [31].

We shall now provide a proof of the general applicability of $H^{top}$ for the
topological classification, and we shall show that the topological invariant
calculated from $H^{top}$ for a system with chiral symmetry separates into
contributions given by an invariant of the substrate and of the in-gap bands
separately. To demonstrate the applicability it is sufficient to demonstrate
that the topological Hamiltonian possesses the same global symmetries as the
full 2D Hamiltonian. Let $\mathcal{H}(k_{x})$ be the full Hamiltonian at a
fixed $k_{x}$ and consider it as a matrix that acts on all other degrees of
freedom including $y$. The Green’s function is defined by
$G(\omega,k_{x})=[\omega\openone-\mathcal{H}(k_{x})]^{-1}$ and as the
Hamiltonian is gapped, $G(\omega=0,k_{x})=-[\mathcal{H}(k_{x})]^{-1}$ is non-
singular. If $\tilde{\Lambda}$ is a symmetry operator such that
$\tilde{\Lambda}\mathcal{H}(k_{x})\tilde{\Lambda}=\pm\mathcal{H}(\pm k_{x})$
for a given combination of $\pm$ signs, then
$\tilde{\Lambda}G(0,k_{x})\tilde{\Lambda}=\pm G(0,\pm k_{x})$ holds as well
[46]. If $\tilde{\Lambda}$ is global such that
$\tilde{\Lambda}=\Lambda\otimes\openone_{y}$ where $\openone_{y}$ is the
identity for the $y$ coordinate and $\Lambda$ acts on the remaining degrees of
freedom [42], then it commutes with the projection onto $y$ and it follows
that

$\Lambda^{-1}G(\omega=0,k_{x},y,y)\Lambda=\pm G(\omega=0,\pm k_{x},y,y).$ (4)

Therefore the local Green’s function, and consequently $H^{top}(k_{x},y)$
satisfy all the global symmetries of the full Hamiltonian. Further, as proven
in Appendix A, since the interface region is bounded, the following theorem is
true:

The Hamiltonian $\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{V}$ possesses (5) a
global symmetry $\Lambda\otimes\openone_{y}$ $\displaystyle\Leftrightarrow$
$\displaystyle\text{$\Lambda$ is a symmetry of $g(0,k_{x},y)$ and $H_{V}$
individually}.$

It follows then from Eq. (2) that the $T$-matrix also possesses the same
symmetry. This means in any symmetry class, we can define a topological
invariant of the topological Hamiltonian, the substrate Green’s function, the
$T$-matrix and the scattering Hamiltonian separately.

We now specialise our discussion to chiral 1D symmetry classes which is
relevant to magnetic impurity-superconductor interfaces [47, 32, 31]. In these
classes, the Hamiltonian, $\mathcal{H}(k_{x})$, has a unitary Hermitian
symmetry $\mathcal{C}$ such that
$\mathcal{C}\mathcal{H}(k_{x})\mathcal{C}=-\mathcal{H}(k_{x})$. In this case
the appropriate topological invariant is the winding number, $W$, calculated
by writing the topological Hamiltonian in the eigenbasis of the chiral
operator [48]

$\displaystyle H^{top}(k_{x},y)=\begin{pmatrix}0&h(k_{x},y)\\\
h^{\dagger}(k_{x},y)&0\end{pmatrix},$ (6)

such that

$\displaystyle W(y)=\frac{1}{2\pi
i}\int_{-\pi}^{\pi}dk_{x}\partial_{k_{x}}\log\\{\det[h(k_{x},y)]\\}.$ (7)

Here $h(k_{x},y)$ is referred to as the chiral decomposition, with half the
dimension of $H^{top}(k_{x},y)$. Theorem (5) ensures that $g(\omega,k_{x},y)$,
$T(\omega=0,k_{x})$ and $H_{V}$ individually admit simultaneous chiral
decompositions as in Eq. (6). At $y=0$ the topological Hamiltonian becomes
$H^{top}(k_{x},y=0)=-H_{V}T^{-1}(0,k_{x})[g(0,k_{x},0)]^{-1}$. From the chiral
decomposition it follows that the winding number can be written as the sum
(see Appendix B for details)

$W(y=0)=W_{g^{-1}(y=0)}-W_{T^{-1}}+W_{H_{V}},$ (8)

which is the central result of this paper. Here $W_{A}$ represents of the
winding number obtained from Eq. (7) with $h(k_{x},y)$ replaced by the chiral
decomposition of $A=[g(\omega=0,k_{x},y)]^{-1}$, $[T(\omega=0,k_{x})]^{-1}$
and $H_{V}$ respectively.

In the case of a trivial substrate and trivial (including local) scattering,
$W_{g^{-1}(y=0)}=W_{H_{V}}=0$ and therefore

$W(y=0)=-W_{T^{-1}}.$ (9)

The invariant only becomes undefined and changes value when
$\det\\{[T(\omega=0)]^{-1}\\}=0$, matching when the in-gap bands touch.

We can also prove that the correct trivial phases are identified. Let $V$ be
the scattering strength in $H_{V}$. In the limit $V\to 0$, Eq. (8) reduces to
$W(y=0)=W_{g^{-1}(0)}$, which is trivial for a trivial substrate. In the
opposite limit $V\to\infty$, we have $H^{top}\sim H_{V}$ and the in-gap modes
are pushed to infinite energy. This Hamiltonian can no longer wind around the
origin and the phase is trivial.

For a general $H_{V}$ with $k_{x}$ dependence, a gap closure could occur.
These gap closures cause zeros in the $T$-matrix which can change its winding
number [46]. However due to the relative negative sign between the two terms
in Eq. (8), any change caused by the appearance of such a zero is perfectly
compensated by the change in winding number of the scattering Hamiltonian (see
Appendix B). We have therefore proven that in the case of a trivial substrate,
the topological Hamiltonian proposed in [31, 32] identifies the correct
trivial states and is only changed by poles of $T(\omega=0,k_{x})$. As a
consequence, it indeed provides the correct topological classification when
$y$ is tuned to the interface.

The classification in the case of a topologically non-trivial substrate via
the topological Hamiltonian remains outstanding though. The topological
Hamiltonian $H^{top}$ contains a dimensional reduction from a higher $D$
dimensional substrate to a lower $d$ dimensional interface. However, the
allowed values of topological invariants changes between dimensions [48] and
symmetries allowing for a non-trivial system in $D$ dimensions may impose only
trivial states in $d$ dimensions. Therefore $[g(\omega=0,k_{x},y)]^{-1}$ does
not necessarily form a good topological Hamiltonian of the substrate, whereas
a trivial state remains trivial after any dimensional reduction as it remains
always amenable to the atomic limit.

When $y\neq 0$, the form of the winding number becomes (see Appendix B),

$\displaystyle W(y)=2W_{g^{-1}(y)}-W_{T^{-1}}$ (10)
$\displaystyle-\frac{1}{2\pi
i}\int_{-\pi}^{\pi}dk\partial_{k}\log\Bigl{\\{}\det\bigl{[}\openone+\tilde{g}^{-1}(-y)\tilde{T}^{-1\dagger}\tilde{g}^{-1}(y)\tilde{g}(0)\bigr{]}\Bigr{\\}},$

where $\tilde{g}^{-1}(y)$ and $\tilde{T}^{-1}$ are the chiral decompositions
of $[g(\omega=0,k_{x},y)]^{-1}$ and $[T(\omega=0,k_{x})]^{-1}$ respectively.
The final term can be singular and possess non-trivial winding. As we show in
Appendix B, for a trivial substrate, as $|y|\rightarrow\infty$ the zeros of
$G(\omega=0,k_{x},y,y)$ cancel the poles of $T(\omega=0,k_{x})$ and the
winding of the $T$-matrix is perfectly cancelled. Therefore
$H^{top}(|y|\rightarrow\infty)$ is trivial. This was observed in the continuum
model of a spiral magnetic interface embedded into a superconductor in Ref.
[31].

Equation (8) is a general result applicable to any gapped substrate with any
interface that has a chiral symmetry. We recall furthermore that it applies
also to interfaces of larger width, by extending the matrix sizes to include
the $y$ coordinates of the interface too. The key appeal of this
classification method is that it is calculated from analytically tractable
objects, well defined in the system size $L\rightarrow\infty$ limit, removing
any finite size effects. As only the local Green’s function is required, this
approach is applicable to first principals calculations [17, 18, 19, 38]. It
also opens the door to connecting this classification to local response
functions that are experimentally accessible.

Figure 2: (a) Winding Number of $H^{top}(k_{x},y=0)$ for $\Delta=0.1t$ and
$k_{F}=0.65$. Wide black dashed lines show the phase boundaries in Eq. (14).
At $k_{m}=0,\pi$ the model is gapless and changes symmetry class. The fuzzy
pattern nearby arises from unavoidable numerical instabilities near the gap
closures. The minima of the lower phase boundaries are occur at
$k_{m}=k_{F},\pi-k_{F}$. (b) Entanglement spectrum $\zeta$ at $k_{m}=k_{F}$
for $V_{m}$ varying along the dotted line in panel (a), computed for a
$50\times 101$ lattice with an entanglement cut made perpendicular to the
impurity chain splitting the system in two. The background colour indicates
the winding number shown in panel (a). Edge modes, characterised by
$\zeta=1/2$, appear exactly when the winding number of the topological
Hamiltonian is non-trivial.

Application to Magnetic Interface in a Superconductor – To demonstrate the
usefulness of this approach we apply it to classify a tight-binding model as
depicted in Fig. 1 of a chain of spiralling magnetic impurities embedded into
a s-wave superconductor. Similarly to Refs. [8, 21, 23, 30, 31], we consider
classical magnetic impurities aligned along the $x$ direction at $y=0$, with
planar magnetisation $\textbf{M}(x)=V_{m}(\cos(2k_{m}x),\sin(2k_{m}x),0)$,
where $V=V_{m}$ sets the scattering strength and $k_{m}$ the period of the
spiral. Translational symmetry along $x$ can be restored by a gauge
transformation on the electron creation operator
$c^{\dagger}_{x,y,\sigma}\rightarrow e^{ik_{m}\sigma
x}c^{\dagger}_{x,y,\sigma}$, for spin $\sigma=\uparrow,\downarrow=+,-$ [49].
This maps the magnetisation to $\mathbf{M}=V_{m}(1,0,0)$ but spin splits the
bands by the shifts $k_{x}\rightarrow k_{x}+\sigma k_{m}$. In $(k_{x},y)$
space the components of the Hamiltonian
$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{V}$ are then written as

$\displaystyle\mathcal{H}_{0}=\sum_{k_{x},y,\sigma}\bigl{[}-2t\cos(k_{x}+\sigma
k_{m})-\mu\bigr{]}c^{\dagger}_{k_{x},y,\sigma}c_{k_{x},y,\sigma}$
$\displaystyle-t\sum_{k_{x},y,\sigma}\Bigl{[}\bigl{(}c^{\dagger}_{k_{x},y+1,\sigma}c_{k_{x},y,\sigma}+\Delta
c^{\dagger}_{k_{x},y,\uparrow}c^{\dagger}_{-k_{x},y,\downarrow}\bigr{)}+\text{h.c.}\Bigr{]},$
(11)
$\displaystyle\mathcal{H}_{V}=V_{m}\sum_{k_{x}}c^{\dagger}_{k_{x},0,\uparrow}c_{k_{x},0,\downarrow}+\text{h.c.},$
(12)

with hopping integral $t$, s-wave superconductor pairing $\Delta$, and
chemical potential $\mu$. The coordinates $x,y$ label a square lattice with
lattice constant $a=1$. The spiral wave-vector $k_{m}$ is left as a free
parameter, not fixed to a free-energy minimum as in systems with effective
magnetic interactions [50, 51, 52, 53, 9, 54, 55]. We consider a constant
$\Delta$ rather than solving self-consistently for the effect of the
impurities, as this provides an almost identical phase diagram (see Appendix
E) but allows for an analytic solution for the phase boundaries.

For $k_{m}\neq n\pi$ where $n\in\mathbb{Z}$, the model is in the BDI symmetry
class [48], with chiral, particle-hole, time reversal and inversion symmetry
operators [56]

$\displaystyle\mathcal{C}=\tau^{y}\sigma^{x},\ \ \mathcal{P}=i\tau^{x}K,\ \
\mathcal{T}=\tau^{z}\sigma^{x}K,\ \ \mathcal{I}=\tau^{z}\sigma^{x},$ (13)

where $K$ is the complex conjugation, and $\sigma^{i},\tau^{i}$ are the Pauli
matrices for spin and particle-hole degrees of freedom, respectively. The
calculation of the substrate Green’s function, $g(\omega,k_{x},y)$, can be
done as in Ref. [30] from which we obtain $T(\omega,k_{x})$, as shown in
Appendix C. The poles of $T(0,k_{x})$ provide the gap closures of the in-gap
bands that mark the topological phase boundaries. This occurs at the high
symmetry points $k_{x}=0,\pi$ at the scattering strengths (see Appendix D)

$V^{*}_{m,k_{x}}=\pm\left\\{\bigl{[}g^{\uparrow}_{11}(k_{x})\bigr{]}^{2}+\bigl{[}g_{12}^{\uparrow}(k_{x})\bigr{]}^{2}\right\\}^{-1/2},$
(14)

where $g^{\uparrow}_{11}$ and $g^{\uparrow}_{12}$ are the
($\uparrow$,particle)-($\uparrow$,particle) and
($\uparrow$,particle)-($\downarrow$,hole) terms of $g(\omega=0,k_{x},y=0)$. In
the continuum limit, $V^{*}_{m,k_{x}=0}$ reproduces the corresponding result
of Ref. [31]. Gap closures occur only at $k_{x}=0,\pi$, and so the winding
number is restricted to $W=0,\pm 1$. Using Eq. (9) the parity of the winding
number of the topological Hamiltonian for $k_{m}\neq n\pi$ can be written as
[47]

$(-1)^{W(y=0)}=\prod_{k_{x}=0,\pi}\operatorname{sign}\left[(V^{*}_{m,k_{x}})^{-2}-(V_{m})^{-2}\right].$
(15)

Consequently a non-trivial phase is obtained for
$\min(V_{m,k_{x}=0}^{*},V_{m,k_{x}=\pi}^{*})<V_{m}<\max(V_{m,k_{x}=0}^{*},V_{m,k_{x}=\pi}^{*})$.
For $k_{m}=n\pi$ where $n\in\mathbb{Z}$, the spin sectors decouple and the
Hamiltonian block diagonalises into two s-wave Hamiltonians with each block
subject to a local Zeeman shift $V_{m}\sigma^{z}$. Each block is in the
trivial CI class such that the winding number is always zero [48].

The resulting phase diagram is show in Fig. 2(a). To further corroborate our
results, in Fig. 2(b) we show the entanglement spectrum $\tilde{\zeta}$ as
$V_{m}$ is varied along the vertical dashed line in Fig. 2(a) obtained from an
entanglement cut made perpendicular to the chain [57]. In a trivial phase
$\tilde{\zeta}$ is gapped about $1/2$, whereas in a non-trivial phase there is
a doubly degenerate value $\tilde{\zeta}=1/2$ [57]. This matches exactly our
phase diagram.

Figure 3: (a) Winding number of $1-2\Pi^{\mathcal{R}}$, the projector
restricted to the impurity site for $N_{y}=101$, $k_{F}a=0.65$ and
$\Delta=0.1t$. The black dashed lines show the phase boundaries from Eq. (14).
The classification gives the incorrect topological invariant, overshooting the
phase boundary for $k_{m}\approx 0$ and undershooting it for $k_{m}\approx
k_{F}$. As in Fig. 2, the fuzzy pattern $k_{m}a=0,\pi$ arises from numerical
instability near gap closures. (b) Same as panel (a), zoomed on $k_{m}\approx
k_{F}$. The additional orange dash-dotted line shows where the gap of the
projector closes, matching exactly where the winding number changes. (c)
Entanglement spectrum $\zeta$ of the projector $P^{\mathcal{R}}$ reduced to
just to $y=0$, at $k_{x}=0$, $k_{m}=k_{F}$ as $V_{m}$ is varied along the
dotted line in panel (b). The black dashed and orange dash-dotted lines mark
the boundaries as in panel (b). The discontinuity in $\zeta$ (black line) mark
the true phase transition, whereas the erroneous phase transition by the
projector appears where the $\zeta$ gap closes instead (orange line).

Classification of Magnetic Interface Model Via Reduced Projector – The
topological Hamiltonian is a local probe, but being derived from the full
Green’s function it carries the full information of the band structure, both
bulk bands delocalised over the substrate and the in-gap bands localised to
the impurity chain. Indeed Eqs. (8) and (9) involve only $y=0$. However
through $T$ they still involve the extended bulk Green’s function $g$, and it
may be natural to ask how much involvement of the extended substrate states is
actually required. As the in-gap bands are the localised to the interface, it
might be possible that topological properties could be completely reproducible
from the wavefunctions truncated to a small wedge around the scattering
interface. As this cannot be done by averaging the Hamiltonian over the $y$
coordinate [31], we should work directly with the ground state projector. Our
goal is to test under which conditions indeed the long-ranged spatial
information of the substrate is of relevance. To this end we compare the
results from $H^{top}$ with the classification scheme proposed in Ref. [42]
under spatial truncation.

Consider a $D$ dimensional trivial substrate with an embedded $d<D$
dimensional topological model and ground state $\ket{GS}$ [42]. The
classification scheme is based on the ground state projector [58]

$\displaystyle
P_{i,j}^{\alpha,\beta}(k)=\bra{GS}c^{\dagger}_{k,i,\alpha}c_{k,j,\beta}\ket{GS},$
(16)

where $k$ is the $d$ dimensional momentum in the translationally invariant
directions, $i,j$ label spatial degrees of freedom in the non-translationally
invariant directions, and $\alpha,\beta$ label unit cell degrees of freedom.
Spatial degrees of freedom can be traced out by restricting the projector to a
spatial region $\mathcal{R}$ that contains the embedded system, leaving a
reduced projector $P^{\mathcal{R}}(k)$ [59]. This projector has eigenvalues
$\zeta_{i}(k)\in[0,1]$ and eigenvectors $\ket{\zeta_{i}(k)}$. A value
$\zeta_{i}$ near 1 refers to a state mainly supported in $\mathcal{R}$ and a
value near 0 a support mainly outside $\mathcal{R}$. The crossing of some
$\zeta_{i}$ through $\zeta=1/2$ indicates thus a qualitative change of the
entanglement between $\mathcal{R}$ and its complement and the possible
topological implication can be captured through the entanglement projector
[42]

$\displaystyle\Pi^{\mathcal{R}}(k)=\sum_{\zeta_{i}(k)>1/2}\ket{\zeta_{i}(k)}\bra{\zeta_{i}(k)},$
(17)

which is in the same symmetry class as the Hamiltonian [42] and
$1-2\Pi^{\mathcal{R}}(k)$ acts like a spectrally flattened band Hamiltonian
with generally a gap about $0$. Consequently it allows for the standard
extraction of topological invariants [48].

An embedded topological insulator has then been defined to be the minimal
spatial region such that the entanglement projector gives a non-trivial
topological invariant and that any further cut causes the projector to become
gapless [42]. This is because any cut through a topological insulator that
maintains protecting symmetries causes a gapless edge mode to appear in the
energy and entanglement spectrum [60]. We shall show in our magnetic interface
model that, though any generically spatial truncation of the $y$-coordinate
leaves a gapped entanglement spectrum, at certain points in the phase diagram,
the entanglement between $\mathcal{R}$ and its complement can cause a gap
closure in the projector, and hence if used uncritically this approach would
produce an incorrect phase diagram. In this case, the truncated region is not
topologically equivalent to the full embedded insulator at all points in the
phase diagram.

Consider indeed the winding number of $1-2\Pi^{\mathcal{R}}$ for the magnetic
interface model [60], with $\mathcal{R}$ reduced to just one site at $y=0$.
This truncation leaves a generically gapped reduced projector. The result is
shown in Fig. 3(a) and it does not match Fig. 2(a), overshooting the phase
boundary for $k_{m}\approx 0$ and undershooting for $k_{m}\approx k_{F}$. This
error is reproduced for larger lattice sizes and in the entanglement spectrum.
To recover the correct phase diagram $\mathcal{R}$ must be enlarged to contain
$\approx 3$ lattice sites on both sides of the magnetic chain. This error is
particularly significant as $k_{m}=k_{F}$ arises naturally for the ground
state if an effective interaction through the substrate between the magnetic
impurities is taken into account [50, 52, 51, 53, 9].

The projector’s invariant can only change if the projector is discontinuous or
its gap closes. The former occurs when the Hamiltonian’s gap closes and the
ground state reconfigures. The latter is associated with a projector
eigenvalue $\zeta=1/2$. This is a maximally entangled mode, in that it it
maximises the von Neumann Entropy $S=-\zeta\log(\zeta)$ [61]. As shown in Fig.
3(b,c) such modes are responsible for the erroneous phase diagram. These modes
occur as the in-gap wavefunctions are spread over a few sites and have a
strong entanglement between the $y$ and spin-Nambu degrees of freedom [31].
Thus by tracing over part of the wavefunction and only considering the central
site it is possible to generate a maximally entangled state.

This result shows that the ground state projector classification requires
before application a careful checking of the extent of the localised
wavefunctions. Alternatively, every phase transition predicted by this method
would need to be checked as to whether it is due to gap closing in the
Hamiltonian or in the reduced projector. Both could become computationally
costly. This underlines the significance of the role of the substrate in the
dimensional reduction that is captured by a Green’s function based method as
put forward in this paper.

Conclusions – We have shown that the topological classification of 1D
interfaces embedded into a topologically trivial 2D substrate with chiral
symmetry can be performed exactly by the local Green’s function tuned to the
interface. With Eq. (8) we then provided an explicit formula disentangling the
local scattering effects from the substrate, while maintaining the full
spatial dependence of the substrate through the $T$-matrix. We demonstrated
further that the full spatial dependence indeed is crucial for the topological
classification, and care has to be taken when using classification schemes
relying on a spatial truncation. Though we believe extensions into higher
dimensions and non-chiral symmetry classes are possible, the proof of the
applicability of a local Green’s function classification to Pfaffian or Chern
invariants remains outstanding. A proof for Pfaffian invariants would cover 1D
Class D superconductors and would complete the classificationof all non-
trivial 1D symmetry classes [62, 48]. The classification of interfaces
embedded into topologically non-trivial substrates with Green’s function based
topological Hamiltonians also requires further study.

Acknowledgements – The authors thank Joe Winter and Konner McNeillie for
helpful discussions. HMS acknowledges studentship funding from EPSRC under
Grant No.EP/W524505/1. The work presented in this paper is theoretical. No
data were produced, and supporting research data are not required.

##  Appendix A Proof of Theorem (5)

We consider the 2D substrate with the embedded scattering interface, described
by the Hamiltonian
$\mathcal{H}(k_{x})=\mathcal{H}_{0}(k_{x})+\mathcal{H}_{V}(k_{x})$. The
scattering interface is assumed to be concentrated to site $y=0$, which could
be a chain of impurities. Alternatively, as explained in the main text, if the
interface has a larger width we can partition the $y$ coordinates into
supercells of the width of the interface, such that integer $y$ label the
position of the supercells, and the $y$ dependence within each supercell is
taken into account as internal matrix index such as spin or particle-hole
degrees of freedom. For an interface in the form of a chain, the supercell
matches the unit cell. Since $\mathcal{H}_{V}(k_{x})$ is restricted to $y=0$
we can write $\mathcal{H}_{V}(k_{x})=H_{V}(k_{x})\otimes P_{y=0}$, where
$P_{y}$ projects onto supercell $y$.

Let $\\{\ket{y}\\}$ form an orthonormal basis of 1D wavefunctions localised at
cell $y$ such that $\bra{y}\mathcal{A}(k_{x})\ket{y^{\prime}}$ reduces any
operator $\mathcal{A}(k_{x})$ to a matrix of internal degrees of freedom only.
We restrict ourselves to substrates that are fully translationally invariant,
so
$\bra{y}\mathcal{H}_{0}(k_{x})\ket{y^{\prime}}=\bra{y+n}\mathcal{H}_{0}(k_{x})\ket{y^{\prime}+n}$
for all $y,y^{\prime},n$. Consider a global symmetry of the Hamiltonian
$\tilde{\Lambda}$ such that
$\tilde{\Lambda}^{-1}\mathcal{H}(k_{x})\tilde{\Lambda}=\pm\mathcal{H}(\pm
k_{x})$, where the $\pm$ are chosen depending on the type of symmetry. As
$\tilde{\Lambda}$ is global, it acts site-wise [42], i.e. the symmetry is of
the form $\tilde{\Lambda}=\Lambda\otimes\openone_{y}$, where $\Lambda$ is an
operator that acts on the degrees of freedom within a supercell and
$\openone_{y}$ is the identity acting on the $y$ cell coordinate. The action
of the symmetry on the Hamiltonian gives

$\displaystyle\tilde{\Lambda}^{-1}\bigl{[}\mathcal{H}_{0}(k_{x})+\mathcal{H}_{V}(k_{x})\bigr{]}\tilde{\Lambda}=\pm\bigl{[}\mathcal{H}_{0}(\pm
k_{x})+\mathcal{H}_{V}(\pm k_{x})\bigr{]}.$ (18)

As $\mathcal{H}_{V}$ is local to the interface, there exists a $y^{*}>0$ such
that $\bra{y^{\prime}}\mathcal{H}_{V}(k_{x})\ket{y^{\prime\prime}}=0$ for
$|y^{\prime}|,|y^{\prime\prime}|>y^{*}$, but also that
$\bra{y^{\prime}}\tilde{\Lambda}^{-1}\mathcal{H}_{V}(k_{x})\tilde{\Lambda}\ket{y^{\prime\prime}}=0$
as $\tilde{\Lambda}$ acts only locally. Therefore for such
$y^{\prime},y^{\prime\prime}$

$\displaystyle\bra{y^{\prime}}\tilde{\Lambda}^{-1}\bigl{[}\mathcal{H}_{0}(k_{x})+\mathcal{H}_{V}(k_{x})\bigr{]}\tilde{\Lambda}\ket{y^{\prime\prime}}$
$\displaystyle=\pm\bra{y^{\prime}}\bigl{[}\mathcal{H}_{0}(\pm
k_{x})+\mathcal{H}_{V}(\pm k_{x})\bigr{]}\ket{y^{\prime\prime}},$ (19)

reduces to

$\displaystyle\bra{y^{\prime}}\tilde{\Lambda}^{-1}\mathcal{H}_{0}(k_{x})\tilde{\Lambda}\ket{y^{\prime\prime}}=\pm\bra{y^{\prime}}\mathcal{H}_{0}(\pm
k)\ket{y^{\prime\prime}}.$ (20)

Due to the translational invariance of $\mathcal{H}_{0}(k_{x})$ the latter
equality holds for any $y^{\prime},y^{\prime\prime}$, and it follows that

$\displaystyle\tilde{\Lambda}^{-1}\mathcal{H}_{0}(k_{x})\tilde{\Lambda}=\pm\mathcal{H}_{0}(\pm
k).$ (21)

This means that $\tilde{\Lambda}$ is a symmetry for $\mathcal{H}_{0}(k_{x})$
separately too. From Eq. (18) it follows in turn that

$\displaystyle\tilde{\Lambda}^{-1}\mathcal{H}_{V}(k_{x})\tilde{\Lambda}=\pm\mathcal{H}_{V}(\pm
k),$ (22)

so that $\tilde{\Lambda}$ is a symmetry also for $\mathcal{H}_{V}(k_{x})$. It
follows further that $\Lambda$ is a symmetry of $H_{V}(k_{x})$.

Finally from Eq. (4) in the main text, we obtain that if $\tilde{\Lambda}$ is
a symmetry of $\mathcal{H}_{0}(k_{x})$, then $\Lambda$ is a symmetry of
$g(\omega=0,k_{x},y)$.

To prove the converse let us assume that $\Lambda$ is a symmetry of
$g(\omega=0,k_{x},y)$. Then $\tilde{\Lambda}=\Lambda\otimes\openone_{y}$ is a
symmetry of $\mathcal{H}_{0}(k_{x})$ for the translationally invariant
substrate. If furthermore $\Lambda$ is a symmetry of $H_{V}(k_{x})$, then
$\tilde{\Lambda}$ is a symmetry of $\mathcal{H}_{V}(k_{x})=H_{V}(k_{x})\otimes
P_{y=0}$. Therefore from Eq. (18), $\tilde{\Lambda}$ is a symmetry of
$\mathcal{H}(k_{x})$. This concludes the proof of Theorem (5).

##  Appendix B Winding Number of Topological Hamiltonian

The winding number can evaluated by writing the Dyson equation for Green’s
function [Eq. (1) in the main text] as follows

$\displaystyle
G(0,k_{x},y,y^{\prime})=g(0,k_{x},y)T(0,k_{x})g(0,k_{x},-y^{\prime})$
$\displaystyle\times\Bigl{\\{}\openone+\bigl{[}g(0,k_{x},-y^{\prime})\bigr{]}^{-1}\bigl{[}T(0,k_{x})\bigr{]}^{-1}$
$\displaystyle\times\bigl{[}g(0,k_{x},y)\bigr{]}^{-1}g(0,k_{x},0)\Bigr{\\}}.$
(23)

where $\openone$ is the identity operator, the $y$ label supercells as
explained in Sec. A, and the written $G,g$ and $T$ are matrices in all
internal degrees of freedom of a supercell. The inversion of the substrate
Green’s function and the $T$-matrix is possible due to the assumption that
both are gapped apart from at phase boundaries. The topological Hamiltonian
[Eq. (3) in the main text] can then be written as

$\displaystyle
H^{top}(k_{x},y)=-\Bigl{\\{}\openone+\bigl{[}g(0,k_{x},-y)\bigr{]}^{-1}\bigl{[}T(0,k_{x})\bigr{]}^{-1}$
$\displaystyle\times\bigl{[}g(0,k_{x},y)\bigr{]}^{-1}g(0,k_{x},0)\Bigr{\\}}^{-1}$
$\displaystyle\times\bigl{[}g(0,k_{x},-y)\bigr{]}^{-1}\bigl{[}T(0,k_{x})\bigr{]}^{-1}\bigl{[}g(0,k_{x},y)\bigr{]}^{-1}.$
(24)

To simplify the notation we shall omit the index $k_{x}$ in most of the
following expressions.

###  B.1 Case $y=0$

Let us consider $y=0$ and set $g=g(0,k_{x},0)$ and $T=T(0,k_{x})$. The
topological Hamiltonian then becomes

$\displaystyle H^{top}(k_{x})$
$\displaystyle=-\bigl{(}\openone+g^{-1}T^{-1}\bigr{)}^{-1}g^{-1}T^{-1}g^{-1}$
$\displaystyle=-\bigl{[}\openone+g^{-1}\bigl{(}H_{V}^{-1}-g\bigr{)}\bigr{]}^{-1}g^{-1}T^{-1}g^{-1}$
$\displaystyle=-H_{V}T^{-1}g^{-1}.$ (25)

Theorem (5) guarantees that $T$, $g$ and $H_{V}$, and therefore $H^{top}$, are
all chiral symmetric, admit a simultaneous chiral decomposition, and can be
written in the chiral eigenbasis

$\displaystyle H_{V}=\begin{pmatrix}0&h_{V}\\\
h_{V}^{\dagger}&0\end{pmatrix},\quad g^{-1}=\begin{pmatrix}0&\tilde{g}^{-1}\\\
\tilde{g}^{-1\dagger}&0\end{pmatrix},$ $\displaystyle
T^{-1}=\begin{pmatrix}0&\tilde{T}^{-1}\\\
\tilde{T}^{-1\dagger}&0\end{pmatrix},\quad H^{top}=\begin{pmatrix}0&h^{top}\\\
h^{top\dagger}&0\end{pmatrix},\quad$ (26)

where

$\displaystyle h^{top}=-h_{V}\tilde{T}^{-1\dagger}\tilde{g}^{-1}.$ (27)

The minus sign does not affect the winding number which can then be evaluated
as

$\displaystyle W(y=0)=\frac{1}{2\pi
i}\int_{-\pi}^{\pi}dk_{x}\partial_{k_{x}}\log[\det(h^{top})]$
$\displaystyle=\frac{1}{2\pi
i}\int_{-\pi}^{\pi}dk_{x}\partial_{k_{x}}\Bigl{\\{}\log[\det(h_{V})]+\log[\det(\tilde{T}^{-1\dagger})]$
$\displaystyle+\log[\det(\tilde{g}^{-1})]\Bigr{\\}}.$ (28)

Consider a general chiral symmetric operator $A$ with chiral decomposition $a$
and take a change of variable $z=\det(a)$ such that the $k_{x}$ integration
over the first Brillouin zone becomes the $z$ integration over the closed
contour $\mathcal{Z}=\\{\,z=\det(a(k_{x}))\,|\,k_{x}\in[-\pi,\pi)\,\\}$. We
can then write the corresponding winding number in the form

$\displaystyle W_{A}=\int_{-\pi}^{\pi}\frac{dk_{x}}{2\pi
i}\partial_{k_{x}}\log\bigl{\\{}\det\bigl{[}a(k_{x})\bigr{]}\bigr{\\}}=\oint_{\mathcal{Z}}\frac{dz}{2\pi
iz}.$ (29)

From this expression we see that if we take $a\rightarrow a^{\dagger}$,
$\det(a)\rightarrow\det(a)^{*}$ then $\mathcal{Z}$ is inverted about the real
axis and the contour is traversed in the opposite direction. This gives an
overall minus to the winding number. Therefore the winding number of the
topological Hamiltonian becomes

$W(y=0)=W_{g^{-1}}-W_{T^{-1}}+W_{H_{V}}.$ (30)

A locally acting $H_{V}$ has a zero winding number. But general $k_{x}$
dependent $H_{V}(k_{x})$ may lead to $W_{H_{V}}\neq 0$, and thus seemingly to
a topological phase transition without gap closure. However, in such a case
the change in $W_{H_{V}}$ is compensated by a simultaneous change in
$W_{T^{-1}}$, so that a gap closure still remains mandatory. To see this, we
should recall that $-W_{T^{-1}}+W_{H_{V}}$ represents the winding of
$H_{V}T^{-1}=\openone-H_{V}g$, thus of $\det(\openone-
h_{V}\tilde{g}^{\dagger})$. This means that the direct winding from $H_{V}$ is
undone but the zeros of $T^{-1}$ are preserved. In case of doubt indeed the
foolproof expression to be used for the winding number is $(\openone-
h_{V}\tilde{g}^{\dagger})$, whereas the $T$-matrix arises naturally from
scattering theory and remains under normal conditions the object of choice.

###  B.2 Case $y\neq 0$

When $y\neq 0$, the chiral decomposition of the Hamiltonian is

$\displaystyle
h^{top}(y)=-\left\\{\openone+\bigl{[}\tilde{g}(-y)\bigr{]}^{-1}\tilde{T}^{-1\dagger}\bigl{[}\tilde{g}(y)\bigr{]}^{-1}\tilde{g}(0)\right\\}^{-1}$
$\displaystyle\times\bigl{[}\tilde{g}(-y)\bigr{]}^{-1}\tilde{T}^{-1\dagger}\bigl{[}\tilde{g}(y)\bigr{]}^{-1},$
(31)

where $\tilde{g}(y)$ is the chiral decomposition of $g(0,k_{x},y)$. The
winding number is then evaluated as before. We obtain

$\displaystyle W(y)=\int_{-\pi}^{\pi}\frac{dk_{x}}{2\pi i}$
$\displaystyle\times\partial_{k}\Bigl{\\{}-\log\bigl{[}\det\bigl{(}\openone+\bigl{[}\tilde{g}(-y)\bigr{]}^{-1}\tilde{T}^{-1\dagger}\bigl{[}\tilde{g}(y)\bigr{]}^{-1}\tilde{g}(0)\bigr{)}\bigr{]}$
$\displaystyle+\log\bigl{[}\det\bigl{(}\bigl{[}\tilde{g}(-y)\bigr{]}^{-1}\bigr{)}\bigr{]}+\log\bigl{[}\det\bigl{(}\tilde{T}^{-1\dagger}\bigr{)}\bigr{]}$
$\displaystyle+\log\bigl{[}\det\bigl{(}\bigl{[}\tilde{g}(y)\bigr{]}^{-1}\bigr{)}\bigr{]}\Bigr{\\}},$
(32)

which gives Eq. (10) in the main text. If we consider the Lehmann
representation of the Green’s function [43]

$\displaystyle
G(\omega,k_{x})=\sum_{n}\frac{\ket{E_{n}(k_{x})}\bra{E_{n}(k_{x})}}{\omega_{+}-E_{n}(k_{x})},$
(33)

where $E_{n}$ and $\ket{E_{n}}$ are the eigenvalues and eigenmodes of the
Hamiltonian and $\omega_{+}=\omega+i0^{+}$, the local Green’s function can be
written as

$\displaystyle G(\omega,k_{x},y,y)=\bra{y}G(\omega,k_{x})\ket{y}$
$\displaystyle=\sum_{n}\frac{\braket{y}{E_{n}(k_{x})}\braket{E_{n}(k_{x})}{y}}{\omega_{+}-E_{n}(k_{x})}.$
(34)

For a trivial substrate the in-gap wavefunctions are localised to the
interface region and normalisable. Therefore for such wavefunctions
$\lim_{|y|\rightarrow\infty}\braket{y}{E_{in-gap}(k_{x})}=0$. Taking thus
$|y|\to\infty$ for generic $V_{m},k_{m}$ suppresses any pole from the in-gap
modes. The remaining poles are from substrate and therefore gapped as the
substrate is trivial. This then means that $W(|y|\rightarrow\infty)=0$ and
$H^{top}(k_{x},|y|\rightarrow\infty)$ is trivial. Examining the form of the
winding number we obtain

$W_{T^{-1}}=-\lim_{|y|\rightarrow\infty}\int_{-\pi}^{\pi}\frac{dk_{x}}{2\pi
i}\\\
\times\partial_{k}\log\Bigl{\\{}\det\Bigl{[}\openone+\bigl{[}\tilde{g}(-y)\big{]}^{-1}\tilde{T}^{-1\dagger}\bigl{[}\tilde{g}(y)\big{]}^{-1}\tilde{g}(0)\Bigr{]}\Bigr{\\}}.$
(35)

From the form of the Green’s function in Eq. (23) we see the term that gives
the $y$ dependent winding number does not have any poles (as we have assumed
the substrate is trivial). As such any change in the winding number as $y$ is
varied must be due to zeros in the determinant of this term and so of
$G(\omega=0,k_{x},y,y)$.

##  Appendix C Fourier transform of lattice s-wave superconductor Green’s
function

The partially Fourier transformed retarded Green’s function of a square
lattice s-wave superconductor is evaluated. The Hamiltonian of such a 2D
superconductor, after taking the gauge transformation
$c^{\dagger}_{x,y,\sigma}\rightarrow
e^{ik_{m}\sigma}c^{\dagger}_{x,y,\sigma}$, where $k_{m}$ is the spiral
wavevector of the magnetic impurity interface, is given by

$\displaystyle\mathcal{H}_{0}(k_{x},k_{y})=\mathcal{H}^{\uparrow}\sigma^{\uparrow}+\mathcal{H}^{\downarrow}\sigma^{\downarrow},$
(36)

where we define $\sigma^{\uparrow,\downarrow}=(\openone\pm\sigma^{z})/2$, for
$\sigma^{z}$ the Pauli-$z$ matrix acting on the spin degrees of freedom and

$\displaystyle\mathcal{H}^{\sigma}=\epsilon_{k_{x},k_{y},\sigma}\tau^{z}+\sigma\Delta\tau^{x},$
(37)

where $\epsilon_{k_{x},k_{y},\sigma}=-2t(\cos(k_{x}+\sigma
k_{m})+\cos(k_{y}))-\mu$, the $\tau^{i}$ are the Pauli matrices acting on the
particle-hole degrees of freedom, $t$ is the hopping integral, $\mu$ the
chemical potential, and $\Delta$ the superconductor pairing amplitude chosen
to be real and positive. The retarded Green’s function of the superconductor
is given by [43]

$g(\omega,k_{x},k_{y})=[\omega_{+}-\mathcal{H}_{0}(k_{x},k_{y})]^{-1},$ (38)

where $\omega_{+}=\omega+i0^{+}$. The partial Fourier transform of the
retarded Green’s function is defined as [30]

$g(\omega,k_{x},y)=\int_{-\pi}^{\pi}\frac{dk_{y}}{2\pi}e^{ik_{y}y}g(\omega,k_{x},k_{y}).$
(39)

For the given Hamiltonian we obtain

$\displaystyle
g(\omega,k_{x},y)=g^{\uparrow}(\omega,k_{x},y)\sigma^{\uparrow}+g^{\downarrow}(\omega,k_{x},y)\sigma^{\downarrow},$
(40)

with

$\displaystyle
g^{\sigma}(\omega,k_{x},y)=\frac{1}{2\pi}\int_{-\pi}^{\pi}dk_{y}\frac{e^{ik_{y}y}(\omega_{+}+\epsilon_{k,\sigma}\tau^{z}+\sigma\Delta\tau^{x})}{\omega_{+}^{2}-\epsilon_{k,\sigma}^{2}-\Delta^{2}}.$
(41)

We should notice that since $g(\omega,k_{x},k_{y})=g(\omega,k_{x},-k_{y})$
this integral is independent of the sign of $y$ and we can thus replace
$y\to|y|$. We then change the integration variable to $z=e^{ik_{y}}$, with
$dz=izdk_{y}$, such that the integration is over the unit circle $S^{1}$ in
the positive, anticlockwise direction, and obtain

$g^{\sigma}(\omega,k_{x},y)=\frac{1}{2\pi
i}\oint_{S^{1}}dz\,z^{|y|-1}\,g(\omega,k_{x},z).$ (42)

As we focus on the in-gap states with $|\omega|<\Delta$, the small $+i0^{+}$
is of no importance we shall use $\omega$ instead of $\omega_{+}$ henceforth.
Focusing on just the denominator in the integrand

$\displaystyle z(\omega^{2}-\epsilon^{2}_{k_{x},z,\sigma}-\Delta^{2})$
$\displaystyle=z\Bigl{\\{}\omega^{2}-\bigr{[}-t(z+1/z)-\mu_{k_{x}}\bigr{]}^{2}-\Delta^{2}\Bigr{\\}}$
$\displaystyle=z(\omega^{2}-\Delta^{2})-\frac{z}{z^{2}}\bigl{[}t(z^{2}+1)+\mu_{k_{x}}z\bigr{]}^{2}$
$\displaystyle=\frac{t^{2}}{z}\Bigl{\\{}z^{2}(\tilde{\omega}^{2}-\tilde{\Delta}^{2})-\bigl{[}(z^{2}+1)+\tilde{\mu}_{k_{x}}z\bigr{]}^{2}\Bigr{\\}},$
(43)

where we have defined $\tilde{\mu}_{k_{x}}=\tilde{\mu}+2\cos(k_{x}+\sigma
k_{m})$ and introduced the dimensionless energies $\tilde{\omega}=\omega/t$,
$\tilde{\Delta}=\Delta/t$, and $\tilde{\mu}=\mu/t$. The integral then becomes

$\displaystyle g^{\sigma}(\omega,k_{x},y)=$ $\displaystyle\frac{1}{2\pi
it^{2}}\oint_{S^{1}}\frac{dzz^{|y|+1}g^{\sigma}(\omega,k_{x},z)}{z^{2}(\tilde{\omega}^{2}-\tilde{\Delta}^{2})-[(z^{2}+1)+\tilde{\mu}_{k_{x}}z]^{2}}.$
(44)

The numerator is given by

$z^{|y|+1}g^{\sigma}(\omega,k_{x},z)=\\\
tz^{|y|}\left[\tilde{\omega}z-(\tilde{\mu}_{k_{x}}z+z^{2}+1)\tau^{z}+z\sigma\tilde{\Delta}\tau^{x}\right].$
(45)

All the poles are contained within the denominator and we can safely proceed
by solving

$\displaystyle
0=z^{2}(\tilde{\omega}^{2}-\tilde{\Delta}^{2})-\bigl{[}(z^{2}+1)+\tilde{\mu}_{k_{x}}z\bigr{]}^{2},$
(46)

or

$\displaystyle
z^{2}(\tilde{\omega}^{2}-\tilde{\Delta}^{2})=\bigl{[}(z^{2}+1)+\tilde{\mu}_{k_{x}}z\bigr{]}^{2},$
(47)

from which the square root leads to the two quadratic equations

$\displaystyle
0=z^{2}+z(\tilde{\mu}_{k_{x}}\pm\sqrt{\tilde{\omega}^{2}-\tilde{\Delta}^{2}})+1.$
(48)

We thus obtain the four roots of the denominator

$z_{\pm_{1},\pm_{2}}=\frac{1}{2}\Bigl{[}\tilde{\mu}_{k_{x}}\pm_{1}\sqrt{\tilde{\omega}^{2}-\tilde{\Delta}^{2}}\Bigr{]}\\\
\pm_{2}\frac{1}{2}\sqrt{\left(-\tilde{\mu}_{k_{x}}\pm_{1}\sqrt{\tilde{\omega}^{2}-\tilde{\Delta}^{2}}\right)^{2}-4}.$
(49)

Then the integral is

$\displaystyle g^{\sigma}(\omega,k_{x},y)=$ $\displaystyle\frac{-1}{2\pi
it}\oint_{S^{1}}\frac{z^{|y|}dz}{(z-z_{+,+})(z-z_{-,+})(z-z_{+,-})(z-z_{-,-})}$
$\displaystyle\times\left[\tilde{\omega}z-(\tilde{\mu}_{k_{x}}z+z^{2}+1)\tau^{z}+z\sigma\tilde{\Delta}\tau^{x}\right].$
(50)

The total minus sign arises from the negative $z^{4}$ coefficient in the
denominator. The integral can then be evaluated using Cauchy’s residue
theorem.

Figure 4: Modulus (black, solid) and phase (red, dashed) of the poles obtained
from Eq. (48) for $\omega=0$ and $\Delta=0.1t$. The black dotted line shows
$|z|=1$. One can verify that as long as $|\omega|<\Delta$, the two poles
$z_{\pm,\operatorname{sign}(\mu_{k_{x}})}$ are within in the unit circle.

In Fig. 4 we show the value of the poles for the example of $\omega=0$ as a
function of $\mu_{k_{x}}$. For $|\omega|<\Delta$ the poles
$z_{1,2}=z_{\pm,\operatorname{sign}(\mu_{k})}$ lie within the unit circle, and
the poles $z_{3,4}=z_{\pm,-\operatorname{sign}(\mu_{k})}$ are outside the unit
circle. The Green’s function then evaluates to

$\displaystyle
g^{\sigma}(\omega,k_{x},y)=\frac{-\xi_{+}(\tilde{\omega}+\sigma\tilde{\Delta}\tau^{x})+i\xi_{-}\sqrt{\tilde{\Delta}^{2}-\tilde{\omega}^{2}}\tau^{z}}{t(z_{1}-z_{2})}$
(51)

where

$\displaystyle\xi_{\pm}=\frac{z_{1}^{|y|+1}}{(z_{1}-z_{3})(z_{1}-z_{4})}\pm\frac{z_{2}^{|y|+1}}{(z_{2}-z_{3})(z_{2}-z_{4})}.$
(52)

We note that $\xi_{+}$ is real and $\xi_{-}$ purely imaginary as
$z_{1}=z_{2}^{*}$ and $z_{3}=z_{4}^{*}$. Consequently $g^{\sigma}$ is real as
expected for a Green’s function when $\omega$ lies in a range with vanishing
density of states. The $\xi_{\pm}$ functions are non-singular as long as
$\Delta\neq 0$. We emphasise that this is the exact Green’s function of a
square lattice tight binding s-wave superconductor, and no low-energy
approximation has been made.

Figure 5: Results of self-consistent calculations for $N_{y}=101$,
$\Delta_{B}=0.1t$ and $k_{F}=0.65$. Panel (a) Shows $\Delta(y)$ for
$k_{m}=k_{F}$ and various values of $V_{m}$. Although the scattering potential
has a considerable effect at the impurity site at $y=0$, the renormalisation
of $\Delta(y)$ almost immediately drops to below 10% independently of the
scattering strength. Panels (b) and (c) show a comparison of the band
structure of the self-consistent solution (black lines) and lowest two
eigenvalues of the non-self consistent $\Delta(y)=\Delta_{B}$ (blue circles)
for $k_{m}=0.63$, and $V_{m}=0.6$ and $4.8$ respectively. There is very little
difference between the two, particularly around the gap closures at
$k_{x}=0,\pi$. The insets confirm that there is little change in the
electronic structure. Panel (d) shows the phase diagram obtained from Eq. (65)
together with the analytically calculated phase boundaries from Eq. (14).
There is almost perfect agreement.

##  Appendix D Phase Diagram

With the explicit form of the Green’s function for the impurity chain given by
Eq. (1) in the main text, we can solve for the conditions of a phase boundary
by finding poles of the $T$-matrix at $\omega=0$,

$\det\bigl{[}T^{-1}(0,k_{x})\bigr{]}=0.$ (53)

This equation can be simplified by making use the model’s chiral symmetry at
$\omega=0$. We can write the $T$-matrix in the eigenbasis of the chiral
symmetry, using the unitary

$U=\frac{1}{\sqrt{2}}\begin{pmatrix}1&0&1&0\\\ 0&i&0&i\\\ 0&-1&0&1\\\
i&0&-i&0\\\ \end{pmatrix},$ (54)

written in the basis

$(c_{k_{x},y=0,\uparrow},c_{k_{x},y=0,\downarrow},c^{\dagger}_{-k_{x},y=0,\uparrow},c^{\dagger}_{-k_{x},y=0,\downarrow}).$
(55)

This transforms the $T$-matrix to

$U^{\dagger}T^{-1}U=\begin{pmatrix}0&\tilde{T}^{-1}\\\
\tilde{T}^{-1\dagger}&0\end{pmatrix},$ (56)

with the notation $T=T(\omega=0,k_{x})$ and

$\det(T^{-1})=|\det(\tilde{T}^{-1})|^{2}.$ (57)

Defining the notation $g_{ij}^{\sigma}=[g^{\sigma}(\omega=0,k_{x},y=0)]_{ij}$
and using that at $\omega=0$ we have $g^{\sigma}_{11}=-g^{\sigma}_{22}$ and
$g^{\sigma}_{12}=g^{\sigma}_{21}$, we obtain

$\tilde{T}^{-1}=\begin{pmatrix}ig^{\uparrow}_{12}-g^{\uparrow}_{11}&\frac{i}{V_{m}}\\\
-\frac{i}{V_{m}}&ig^{\downarrow}_{12}-g_{11}^{\downarrow}\end{pmatrix}.$ (58)

The condition for a phase transition is $\det(T^{-1})=0$, which becomes

$\displaystyle
0=g^{\uparrow}_{11}g^{\downarrow}_{11}-g_{12}^{\uparrow}g_{12}^{\downarrow}-i(g_{11}^{\uparrow}g_{12}^{\downarrow}+g^{\downarrow}_{11}g^{\uparrow}_{12})-\frac{1}{V_{m}^{2}}.$
(59)

The bare Green’s function is real as the density of states is zero at
$\omega=0$, and as noted earlier we can omit the infinitesimal shifts
$i0^{+}$. Then the real and imaginary parts split into two equations. The
imaginary part gives

$g^{\downarrow}_{11}g^{\uparrow}_{12}=-g^{\uparrow}_{11}g^{\downarrow}_{12}.$
(60)

If $k_{m}\neq n\pi$ for $n\in\mathbb{Z}$, this has solutions for any set of
system parameters, but required $k_{x}=0,\pi$. This is because the only
differences between the two spin species is the kinetic energy
$\cos(k_{x}+\sigma k_{m})$ and the sign of the pairing $\sigma\Delta$. At
$k_{x}=0,\pi$, the dispersions are equal as $\cos(k_{x}+\sigma
k_{m})=\pm\cos(\sigma k_{m})=\pm\cos(k_{m})$. Therefore
$g^{\uparrow}_{11}=g^{\downarrow}_{11}$ and
$g^{\uparrow}_{12}=-g^{\downarrow}_{12}$, and so the above equation is
satisfied. When $k_{m}=0,\pi$, the above equation is satisfied for any $k_{x}$
meaning the gap closes and stays closed rather than opening up again.

The real part gives the conditions for the phase transition and along with the
symmetry of the spin species at $k_{x}=0,\pi$. We find

$\displaystyle
V^{*}_{m,k_{x}}=\pm\left\\{[g^{\uparrow}_{11}(k_{x})]^{2}+[g_{12}^{\uparrow}(k_{x})]^{2}\right\\}^{-1/2}.$
(61)

giving Eq. (14) in the main text. As gap closures only occur at $k_{x}=0,\pi$,
the winding number is restricted to $W=0,1,-1$ and therefore it is sufficient
to find its parity. This can written as the sign of the product of the
determinants of the chiral decomposition at $k_{x}=0,\pi$ [47]. Using the
chiral decomposition above, this is given by

$\displaystyle(-1)^{W(y=0)}=\prod_{k_{x}=0,\pi}\operatorname{sign}\left[({g^{\uparrow}_{11}})^{2}+({g^{\uparrow}_{12}})^{2}-({V}_{m})^{-2}\right]$
$\displaystyle=\prod_{k_{x}=0,\pi}\operatorname{sign}\left[(V^{*}_{m,k_{x}})^{-2}-(V_{m})^{-2}\right],$
(62)

giving Eq. (15) in the main text.

##  Appendix E Phase Diagram of Self-Consistent Solution

We describe a self-consistent solution to the superconducting gap equations in
the presence of a magnetic interface and show that it makes very little
difference to the overall phase diagram. The self-consistency condition is the
same that was used in Ref. [30, 31]. We restrict ourselves to just s-wave
pairing as the triplet pairing is extremely weak. We maintain translational
invariance along the $x$-direction. Then the self consistency condition is

$\displaystyle\Delta(y)=V_{P}\braket{c^{\dagger}_{x,y,\uparrow}c^{\dagger}_{x,y,\downarrow}}=V_{P}\sum_{k_{x}}\braket{c^{\dagger}_{k_{x},y,\uparrow}c^{\dagger}_{-k_{x},y,\downarrow}},$
(63)

where $\braket{\dots}$ denotes the ground state average. The translational
invariance along the $x$-direction means the pairing is only a function of
$y$. The pairing potential strength $V_{P}$ is chosen such that the pairing
tends to the chosen bulk pairing, $\Delta_{B}$ (i.e. the pairing at $V_{m}=0$)

$V_{P}=\frac{\Delta_{B}}{\sum_{k_{x}}\braket{c^{\dagger}_{k_{x},}{y}{\rightarrow\infty,\uparrow}c^{\dagger}_{-k_{x},|y|\rightarrow\infty,\downarrow}}{missing}{missing}}.$
(64)

Starting from the initial guess $\Delta(y)=\Delta_{B}$ we calculate
$\braket{c^{\dagger}_{k_{x},y,\uparrow}c^{\dagger}_{-k_{x},y,\downarrow}}$
from direct diagonalisation of the tight-binding matrix. With this result we
update $V_{P}$ through Eq. (64) to guarantee that $\Delta(y)$ tends to
$\Delta_{B}$ at large $y$. With this we reconstruct the Hamiltonian and
iterate until convergence. With the final solution of the tight-binding model
we calculate a topological invariant of the bands. As it is computationally
heavy to calculate the winding number of a large matrix, we make use of the
inversion symmetry to calculate an equivalent invariant. This is defined as

$\nu=n_{0}-n_{\pi},$ (65)

where $n_{k_{x}}$ is the number of negative non-zero eigenvalues of
$P(k_{x})\mathcal{I}P(k_{x})$ where $\mathcal{I}$ is the inversion symmetry
operator. This has been shown to be equal to the number of $\zeta=1/2$ modes
in the entanglement spectrum localised to the edge of the system [57].

The self consistent solution to the gap is shown in Fig. 5(a) for various
$V_{m}$. We see that the gap is lowered by around 20% around the magnetic
impurity around the critical scattering $V_{m,k_{x}=0}^{*}\approx 0.63t$. The
residual error of this solution is on the order of $10^{-8}\Delta_{B}$. We
show the band structure for two values of $V_{m}$ around the phase boundaries
in Fig. 5(b)–(c) and compare it to the in-gap bands of the non-self-consistent
solution. We again see that around the bottom of the band very little
difference is made and the only visible deviations are around the
$k_{x}\approx 2k_{F}$ where the in-gap bands of the self-consistent solution
are slightly lower than the non-self-consistent one. Finally, in Fig. 5(d) we
show the phase diagram of the invariant $\nu$ defined above with the non-self-
consistent phase boundaries overlayed and again we see perfectly agreement up
to the pixel size. We conclude that self-consistency has so little effect on
all relevant results such that it can be safely ignored.

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Also at ]Kazan Federal University, Kazan 420008, Russia

# Multi-step particle emission probabilities in superheavy nuclei at moderate
excitation energies

A. Rahmatinejad  Joint Institute for Nuclear Research, Dubna, 141980, Russia
T. M. Shneidman  [  Joint Institute for Nuclear Research, Dubna, 141980,
Russia  G. G. Adamian<EMAIL_ADDRESS>Joint Institute for Nuclear
Research, Dubna, 141980, Russia  N. V. Antonenko  Joint Institute for Nuclear
Research, Dubna, 141980, Russia and Tomsk Polytechnic University, 634050
Tomsk, Russia  P. Jachimowicz  Institute of Physics, University of Zielona
Góra, Szafrana 4a, 65516 Zielona Góra, Poland  M. Kowal  National Centre for
Nuclear Research, Pasteura 7, 02-093 Warsaw, Poland

###### Abstract

The probabilities of $xn$-, $pxn$-, and $\alpha xn$-evaporation channels in
excited superheavy nuclei were evaluated using the Monte Carlo method. The
calculations utilized microscopically determined nuclear level densities and
were compared with results obtained from the phenomenological Jackson formula.
Effective temperatures derived from the microscopic approach were incorporated
into the Jackson formula for different evaporation channels at low and
moderate excitation energies. Additionally, an analytical formula was
introduced to estimate the average kinetic energy of emitted particles in
multi-step processes.

Superheavy nuclei, microscopic-macroscopic model, fission.

###### pacs:

21.10.Ma, 21.10.Pc, 24.60.Dr, 24.75.+i

Superheavy nuclei (SHN) such as Cn, Nh, Fl, Mc, Lv, Ts, and Og, with charge
numbers ranging from 112 to 118, have been successfully synthesized through
complete fusion reactions induced by 48Ca projectiles and actinide targets.
These reactions occur in the $xn$-evaporation channels, where $x$ represents
the number of emitted neutrons. The literature sources [1, 2, 3, 4, 5, 6, 7,
8, 9] provide detailed information on these reactions.

Most of these superheavy nuclei have been produced in the $3n$\- and
$4n$-evaporation channels, with excitation energies of the compound nucleus
(CN) in the $E_{CN}=30-40$ MeV. This achievement raises the question of
expanding the table of nuclides even further. One possibility is the operation
of the Factory of SHN at the Joint Institute for Nuclear Research (Dubna) [5,
10], which could offer a new avenue for studying and producing new isotopes of
superheavy nuclei at moderate excitation energies ($E_{CN}\geq 50$ MeV).

Nevertheless, the rate at which the survival probability of the CN and,
consequently, the cross-section for evaporation residues decrease with the
augmentation of beam energy remains a subject of ongoing inquiry [11, 12].

One of the main ingredients in calculating the survival probability

$\displaystyle
W_{s}(U)=P(s,U)\prod_{i_{s}=1}^{x_{s}}\frac{\Gamma_{i_{s}}(U_{i_{s}})}{\Gamma_{t}(U_{i_{s}})}$
(1)

is the probability $P(s,U)$ of a particle evaporation channel [a
$xn$-evaporation channel ($s=xn$, $x_{s}=x$) or a channel with the charged
particle ($p$ or $\alpha$) emission followed by $xn$-evaporation ($s=pxn$ or
$\alpha xn$, $x_{s}=x+1$)] at the excitation energy $U$ [13]. In Eq. (1), the
total width $\Gamma_{t}$ is the sum of widths of particle evaporation
$\Gamma_{i_{s}}$ and fission $\Gamma_{f}$. The widths are calculated using the
corresponding thresholds for decays and level densities [14, 15, 16, 17, 18,
19]. The probability $P(xn,U)$, from which the positions of the excitation
function maxima crucially depend, is calculated in the literature using the
Jackson analytic formula [20]

$\displaystyle P(xn,U)=I(\Delta_{x},2x-3)-I(\Delta_{x+1},2x-1),$
$\displaystyle
I(\Delta_{x},2x-3)=1-\exp[-\Delta_{x}]\left[\sum_{i=0}^{2x-3}\frac{(\Delta_{x})^{i}}{i!}\right],$
(2)

where $I$ is Pearson’s incomplete gamma function and
$\Delta_{x}=\left(U-B_{xn}\right)/T$, $B_{xn}=\sum_{i=1}^{x}B_{i}$, $T$ is the
effective temperature, and $B_{i}$ is the neutron binding energy [20]. The
first incomplete gamma function gives the probability that the original CN
will emit at least $x$ neutrons. The second function gives the probability
that the residual nucleus has enough excitation energy to emit at least $x+1$
neutrons finally. For the final residual nucleus after emission of $x$
neutrons, with an excitation energy higher than the corresponding fission
barrier height $B_{f}$ but insufficient for neutron evaporation, $B_{x+1}$ is
replaced by $B_{f}$ in the calculation of $\Delta_{x+1}$ [13]. In the case of
exactly one neutron emission, Eq. (Multi-step particle emission probabilities
in superheavy nuclei at moderate excitation energies) is transformed into a
simple formula

$\displaystyle P(1n,U)=\begin{dcases}1,&B_{1n}<U<B_{2n}\\\
e^{-\Delta_{2}}(1+\Delta_{2}),&U\geq B_{2n},\end{dcases}$ (3)

because the probability of emitting at least $1$ neutron for $U>B_{1}$ equals
one.

When calculating $P(s,U)$ for a channel in which a charged particle (for
example, an $\alpha$-particle or proton) evaporates, it is necessary to expand
the expression for $P(s,U)$ taking into account the corresponding Coulomb
barrier when calculating the corresponding value of $\Delta_{x}$ [21].

Equation (Multi-step particle emission probabilities in superheavy nuclei at
moderate excitation energies) is derived by incorporating the three main
assumptions: i) The neutron energy spectrum is given by the Maxwellian
distribution $\varepsilon_{i}e^{-\varepsilon_{i}/T}$, where $\varepsilon_{i}$
is the kinetic energy of the neutron. ii) The neutron is necessarily
evaporated if the nucleus has enough excitation energy. iii) The effective
temperature $T$ is independent of excitation energy. However, the last
assumption becomes inaccurate for $xn$-evaporation channels with $x\gg 1$.

The decrease in nuclear temperature in the evaporation process is taken into
account in the calculations by an effective temperature

$\displaystyle T=T_{CN}/\sqrt{C},$ (4)

where $C$ is a constant and $T_{CN}$ is the temperature of the initial CN
[21].

The primary objective of this letter is to justify the utilization of the
Jackson formula, incorporating the relation (4), while also determining the
parameters $C$ for the $xn$-, $pxn$-, and $\alpha xn$-evaporation channels
($x$ = 1–9) of excited SHN within the framework of a microscopic approach.
Another aim is to investigate the average kinetic energy of emitted particles
at low and moderate excitation energies. This investigation offers valuable
insights for estimating the survival probabilities and, consequently, the
production cross sections of SHN across a wide range of excitation energies
for the original CN. It can also improve the predictions for the cross
sections.

The determination of $P(s,U)$ can be effectively achieved using the Monte
Carlo (MC) method. In a multi-step process, MC sampling necessitates a
probability distribution of particle kinetic energy to assess the feasibility
of ejecting the next particle at each step. We define the probability density
of emitting a particle with kinetic energy within a small interval centered
around $\varepsilon_{i}$ as follows:

$\displaystyle
p(\varepsilon_{i},E_{i})=N\varepsilon_{i}\rho_{res}(E_{i}-B_{i}-\varepsilon_{i})\delta\varepsilon_{i},$
(5)

where $\rho_{res}(E_{i}-B_{i}-\varepsilon_{i})$ is the nuclear level density
(NLD) of the residue nucleus, and $N$ is the normalization constant. Particle
emissions depend strongly on the NLD of the corresponding final states. Also,
when a multi-step process becomes essential at high excitation energies, NLD
is necessary to obtain additional information for calculating $P(s,U)$.
Several methods for calculating NLD are usually based on two approaches:
combinatorial and thermodynamic [23, 22, 24, 25]. In this paper, the
statistical formalism based on the superfluidity model [26, 27, 28, 29, 30,
14, 31] is used to calculate the NLD of nuclei formed after the emission of a
neutron, a proton, and an $\alpha$-particle. This method consistently
considers pairing and shell effects when estimating the probabilities of
$xn$\- and $pxn$ or $\alpha xn$-evaporation channels. The single-particle
energies and decay thresholds are obtained within the microscopic-macroscopic
approach based on the deformed single-particle Woods-Saxon potential and
Yukawa-plus-exponential macroscopic energy [32, 33]. As shown in Ref. [30],
the model describes well the experimental NLD for various Dy and Mo isotopes.
The CN excitation energy

$\displaystyle E_{i}=U-\sum_{j=1}^{i-1}\left(B_{j}+\varepsilon_{j}\right)$ (6)

In the $i^{\textrm{th}}$ evaporation step, the probability of particle
emission depends on the particle emission thresholds $B_{j}$ and the kinetic
energies $\varepsilon_{j}$ carried away by the particles emitted in previous
steps. We iterate the MC sampling process for consecutive decays until the
nucleus depletes the required excitation energy for the subsequent step.
During the calculation, we record and average the kinetic energies of the
particles over the number of samples performed for each subsequent step in the
$s$-evaporation channel.

Figure 1: Calculated probabilities of ($1n$–$9n$)-evaporation channels for the
nucleus 288F as a function of excitation energy $U$. Solid lines are
calculations within the microscopic approach; dashed lines are calculations
using the analytical formula (Multi-step particle emission probabilities in
superheavy nuclei at moderate excitation energies). The effective temperatures
are taken as $T=T_{CN}/\sqrt{2}$ (a) and $T=\sqrt{U_{CN}/(2a_{CN})}$ (b). The
values of $T_{CN}$, $U_{CN}$, and the level density parameter $a_{CN}$ for the
initial CN are microscopically calculated. Figure 2: Probability of $\alpha
xn$-evaporation channel ($x$=0–8) for 296Lv (a), and $pxn$-evaporation channel
($x$=0–8) for 291Mc (b) calculated within the microscopical approach (solid
lines) and with the analytical formula (Multi-step particle emission
probabilities in superheavy nuclei at moderate excitation energies) modified
for charged particle emissions (dashed lines) versus excitation energy $U$.
The effective temperatures are taken as $T=T_{CN}/\sqrt{2.25}$ with the
original CN temperatures $T_{CN}$. Figure 3: For the 296Lv nucleus, the
average kinetic energies carried away by neutrons at $i$=1–4 (a)-(d)
intermediate evaporation steps in the $5n$-evaporation channel, calculated
microscopically (green lines with asterisks) and with the formulas $2T_{i}$
(solid blue lines) and (7) (red dotted lines) depending on the excitation
energy $U$.

In Figs. 1 and 2, the probabilities $P(s,U)$ of $xn$-, $pxn$-, and $\alpha
xn$-evaporation channels ($x$=1–9) in superheavy nuclei 288Fl, 291Mc and 296Lv
at excitation energies of (10 – 100) MeV are calculated by MC method using Eq.
(5) and microscopic NLD. For comparison, $P(s,U)$ are calculated using the
phenomenological Jackson formula (Multi-step particle emission probabilities
in superheavy nuclei at moderate excitation energies). As can be seen for the
288Fl nucleus in $xn-$evaporation channels, the Jackson formula (Multi-step
particle emission probabilities in superheavy nuclei at moderate excitation
energies) with $C=2$ [see Eq. (4)] gives results close to those of microscopic
calculations. The maximum values of $P(xn,U)$ for $x=1-5$ depend weakly on
$C$. However, due to the cumulative effect of temperature in Eq. (Multi-step
particle emission probabilities in superheavy nuclei at moderate excitation
energies) the results deviate slightly from microscopic calculations as $x$
increases. In Fig. 1 (a), the values of initial CN temperature $T_{CN}$ in Eq.
(4) are taken from microscopic calculations. One can also calculate the
effective temperature $T=\sqrt{U_{CN}/(Ca_{CN})}$ ($C=2$) within the Fermi-gas
model, where $U_{CN}$ and $a_{CN}$ are the excitation energy and the level
density parameter of the parent CN, respectively. For the 288Fl nucleus, this
is shown in Fig. 1 (b).

In Fig. 2, we compare the Monte Carlo (MC) calculations of $P(\alpha xn,U)$
for 296Lv and $P(pxn,U)$ for 291Mc, where $x$ ranges from 0 to 8, with the
corresponding values obtained using the modified analytical formula (Multi-
step particle emission probabilities in superheavy nuclei at moderate
excitation energies) for emitting charged particles.

For the $pxn$ or $\alpha xn$-evaporation channels, we find that the Jackson
formula (Multi-step particle emission probabilities in superheavy nuclei at
moderate excitation energies) with a slightly larger constant value of
$C=2.25$ compared to that used for $xn$-evaporation channels provides a good
reproduction of the values of $P(s,U)$ obtained from microscopic calculations.
This can be attributed to the higher energy requirement for the emission of
charged particles in the initial step, resulting in a significant decrease in
temperature compared to neutron emission.

In each evaporation step $i$ with the corresponding nucleus temperature
$T_{i}$, the average kinetic energy of the Maxwellian distribution is equal to
twice the temperature: $\bar{\varepsilon}_{i}=2T_{i}$. For the 296Lv nucleus,
the excitation energy dependence of the average kinetic energy
$\bar{\varepsilon}_{i}$ ($i$=1–4) of the neutrons in the $5n$-evaporation
channel obtained using the MC method is presented in Fig. 3 together with
$\bar{\varepsilon}_{i}=2T_{i}$. From this figure, it can be concluded that
moderate excitation energies support the assumption of the Maxwellian
distribution. However, at low excitation energies, it becomes vital to
consider the emission of neutrons with kinetic energies less than $2T_{i}$ in
each evaporation step for the correct treatment of the channel’s opening. The
expression

$\displaystyle\bar{\varepsilon}_{i}(xn,U)=2T_{i}\left(1-e^{-\frac{U-B_{xn}}{x}}\right)$
(7)

for the average kinetic energy in the $xn$-evaporation channel leads to good
agreement with the MC calculations over the entire energy range. This
expression is examined for 296Lv in Fig. 3 (red dashed lines). The average
kinetic energies for $pxn$ or $\alpha xn$-evaporation channels can be obtained
similarly.

In conclusion, our study demonstrates that the analytical Jackson formula,
with effective temperatures of $T=T_{CN}/\sqrt{2}$ and $T=T_{CN}/\sqrt{2.25}$,
accurately reproduces the probabilities of $xn$, $pxn$, and $\alpha
xn$-evaporation channels in moderately excited SHN, as calculated using the
microscopic model.

Furthermore, we derived an analytical formula (7) for the average kinetic
energies of emitted particles. This expression provides improved results
compared to those obtained using the Maxwellian distribution, particularly at
low excitation energies in intermediate steps of the evaporation chain,
especially for many-particle emissions.

Moreover, our findings indicate that the accuracy of the calculated
probabilities of $xn$, $pxn$, and $\alpha xn$-evaporation channels, as well as
the average kinetic energies of emitted particles, can be enhanced by using
the temperature $T_{CN}$ of the initial CN calculated within the Fermi-gas
model with an energy-dependent level density parameter.

Overall, our results contribute to a better understanding of particle emission
processes in SHN and provide valuable insights for more accurate predictions
of evaporation channels and properties of emitted particle that play a crucial
role in predicting the survival of newly synthesized SHN.

Acknowledgements:

M.K. was co-financed by the National Science Centre under Contract No.
UMO-2013/08/M/ST2/00257 (LEA COPIGAL). T.M.S., G.G.A., and N.V.A. were
supported by Ministry of Science and Higher Education of the Russian
Federation (Moscow, Contract No. 075-10-2020-117).

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