# Microscopic properties of fractional vortices and domain walls in three-band
$s+is$ superconductors

Igor Timoshuk  Department of Physics, The Royal Institute of Technology,
Stockholm SE-10691, Sweden  Wallenberg Initiative Materials Science for
Sustainability, Department of Physics, The Royal Institute of Technology,
Stockholm SE-10691, Sweden  Egor Babaev  Department of Physics, The Royal
Institute of Technology, Stockholm SE-10691, Sweden  Wallenberg Initiative
Materials Science for Sustainability, Department of Physics, The Royal
Institute of Technology, Stockholm SE-10691, Sweden

###### Abstract

We report microscopic solutions for vortices carrying a variable fraction of
magnetic flux quantum and domain walls in a three-band $s+is$ superconductor
and investigate their properties. The solutions are obtained in a fully self-
consistent treatment of the three-band Bogoliubov-de-Gennes model.

##  I Introduction

The flux quantization is a fundamental property of conventional
superconductors [1, 2]. In London’s argument, the flux quantization is a
consequence of the quantization of the circulation of the $2\pi$-periodic
phase variable, called the phase winding $\oint\nabla\theta=2\pi N$ where $N$
is an integer. At a deeper level, the existence of the phase comes as a
consequence of the concept that a superconductor is a state of matter that
spontaneously breaks $U(1)$ gauge symmetry and, hence, is described by a
complex order parameter field [3, 4]. These circumstances ensure that in an
infinite sample, there are no stable solutions violating the quantization
111Note that no such penalty exists in mesoscopic samples. Also, layered
systems and systems with phase jumps at grain boundaries and weak links have
geometric effects that can affect enclosed flux; here, we do not discuss such
geometric effects. Similar symmetry-based arguments were used to predict that
chiral $p$-wave superconductors allow half-quantum vortices [6, 7]. A
different proposal was made in [8], where it was suggested that in contrast to
symmetry-based arguments, vorticity could exist in a gap field of an
individual band of a multiband superconductor. In that case, a vortex can
carry an arbitrary fraction of magnetic flux quantum (recently, such vortices
were termed unquantized vortices). A gap field in an individual band is not an
order parameter in Landau’s theory sense. Namely, adhering to symmetry-based
Landau theory would imply only one order parameter since only one symmetry is
broken. Hence, it would imply the presence of only one phase field, precluding
the existence of vortices that carry an arbitrary fraction of magnetic flux
quantum. Namely, in the conventional argument, a classical field — the order
parameter –should describe all macroscopic aspects of motion regardless of
microscopic detail; the number of bands is one of such ostensibly
inconsequential microscopic detail.

On the other hand, a classical field theory concept is more general than an
order parameter concept and does not necessarily require a set of broken
symmetries. The initial constructions of “unquantized” vortex in [8] in a two-
band model with single broken symmetry relied on several assumptions. Firstly,
in the presence of the Josephson coupling between components of a two-
component model, there are no well-defined multiple “Mexican hat” effective
potentials with multiple degenerate valleys. Hence, one cannot define two
phase variables using the usual symmetry principles. Nonetheless, it was
argued, at the level of phenomenological models in [8], that there can be a
“deformed” and “non-degenerate” valley in the landscape of a free energy
functional that still allow the introduction of two classical phase-like
variables and sustain energetically stable windings in the phase variables.
The recent experiment [9] reported the observation of vortices carrying a
temperature-dependent fraction of the flux quantum in Ba1-xKxFe2As2 . Notably,
the experiments demonstrated that the objects with fractional flux (i) can be
created in various locations of the samples, giving consistent flux fractions
irrespective of their position, (ii) vortices are mobile, and their position
can be manipulated (iii) it was possible to create fractional and integer
vortices at the same position, which proves that the observations are not
artifacts of some crystal defects. More recently, vortex core
fractionalization was observed in a related material KFe2As2 [10]. Besides the
fact that the phenomenon is interesting on its own, the vortices should
exhibit anyonic statistics. That is, according to the earlier work [11], a
composite object of arbitrary flux tube and electron is an anyon [11, 12, 13,
14]. Hence, due to vortex core states, the unquantized vortices physically
realize such an object in a superconductor and, hence, should obey fractional
statistics.

The material Ba1-xKxFe2As2 where fractional vortices were observed is a
multiband, spin-singlet superconductor. At the doping value $x\approx 0.77$,
it breaks the time-reversal symmetry [15, 16]. The time-reversal symmetry is
broken above the superconducting phase transition [17, 18], due to fluctuation
effects [19, 20, 21]. This means that there are at least two components in the
order parameters when the system transitions to the diamagnetic state.
However, the situation is more complex even in the case of multiple broken
symmetries in multiband superconductors. By the same argument as discussed
above, more than two phase-like variables are possible in Ba1-xKxFe2As2 ,
despite only two broken symmetries, which implies possibly more than two kinds
of fractional vortices [22, 8, 23]. It is still a subject of current
experimental research how many types of fractional vortices can exist in
Ba1-xKxFe2As2 .

Since the existence of “unquantized vortices” cannot be guaranteed by
symmetry, and instead, one relies on interaction to produce a particular
energy landscape, a detailed microscopic investigation of an isolated vortex
solution is warranted. We present such solutions in a fully self-consistent
microscopic model, including intercomponent gauge-field coupling. A similar
solution was obtained in the Bogoliubov- de Gennes (BdG) model in [9], but
only a few characteristics were studied. This work reports a detailed
numerical investigation of the unquantized vortices and their properties.

##  II Model

We focus on a three-band model where the interband Josephson coupling ensures
that the model has only a single local $U(1)$ symmetry spontaneously broken.
We define this model on a two-dimensional square lattice, described by the
microscopic Hamiltonian

$\displaystyle H=$
$\displaystyle-\sum_{\alpha\sigma}\sum_{<ij>}\exp{\mathrm{i}qA_{ij}}c^{\dagger}_{i\sigma\alpha}c_{j\sigma\alpha}$
(1)
$\displaystyle-\sum_{i\alpha\beta}V_{\alpha\beta}c_{i\uparrow\alpha}^{\dagger}c_{i\downarrow\alpha}^{\dagger}c_{i\downarrow\beta}c_{i\uparrow\beta}\,.$

Here $<ij>$ denotes nearest neighbor pairs, and $c_{i\sigma\alpha}$ is the
fermionic annihilation operator at position $i$, with spin $\sigma$
($\sigma\in\\{\uparrow,\downarrow\\}$) and in band $\alpha$
($\alpha\in\\{1,2,3\\}$), phase factor $\exp{\mathrm{i}qA_{ij}}$ accounts for
interaction with the magnetic vector potential
$A_{ij}=\int_{j}^{i}\bm{A}\cdot\differential{\bm{\ell}}$ through Peierls
substitution [24, 25]. The quartic interaction term, defined by
$V_{\alpha\beta}=V_{\beta\alpha}^{*}$, allows Cooper pairs to form and tunnel
between bands. Inter-band coupling dictates no extra $U(1)$ degeneracies. We
consider repulsive interband Josephson coupling so that the model also breaks
time-reversal symmetry [26, 27] to make the connection with Ba1-xKxFe2As2 [15,
16]. The model has two broken symmetries but three components. Hence, three
types of fractional vortices in this model cannot be justified by the standard
“ground state manifold”-based topological classification of defects.

By performing the mean field approximation in the Cooper channel, we obtain
the mean-field Hamiltonian

$\displaystyle\begin{gathered}\mathcal{H}=-\sum_{\sigma\alpha}\sum_{<ij>}\exp{\mathrm{i}qA_{ij}}c^{\dagger}_{i\sigma\alpha}c_{j\sigma\alpha}+\\\
+\sum_{i\alpha}\left(\Delta_{i\alpha}c^{\dagger}_{\uparrow
i\alpha}c^{\dagger}_{\downarrow
i\alpha}+{\rm{H.c.}}\right)+\frac{1}{2}\sum_{\rm{plaquettes}}B_{z}^{2}\,,\end{gathered}$
(4)
$\displaystyle\Delta_{i\alpha}=\sum_{\beta}V_{\alpha\beta}\expectationvalue{c_{\uparrow
i\beta}c_{\downarrow i\beta}}\,,$ (5)

where ${\rm{H.c.}}$ denotes Hermitian conjugation. Discrete version of
Maxwell’s equation $\nabla\times\nabla\times\bm{A}=\bm{J}$ determines the
connection between $A_{ij}$ and $J_{ij}$

$J_{ij}=-2q\sum_{\alpha\sigma}\Im{\expectationvalue{c_{i\sigma\alpha}^{\dagger}c_{j\sigma\alpha}}\exp{\mathrm{i}qA_{ij}}}.$
(6)

The free energy for the system Eq. (4) can be calculated as

$\begin{gathered}F_{H}=\sum_{i}\bm{\Delta}_{i}^{\dagger}V^{-1}\bm{\Delta}_{i}-k_{\rm{B}}T\Tr\ln\quantity(\mathrm{e}^{-\beta\mathcal{H}}+1)+\\\
+\frac{1}{2}\sum_{\mathrm{plaquettes}}B^{2},\end{gathered}$ (7)

where the magnetic field $B=\nabla\times\bm{A}$ is defined on plaquettes.

Using the iteration scheme, described in [23], the solution for equations Eq.
(5), Eq. (6), along with the Maxwell equation, are obtained. We used the
approximate Chebyshev spectral expansion method. The self-consistent iteration
procedure stops when the convergence criteria $\absolutevalue{\delta
p/\left(p+\epsilon\right)}<\varepsilon$ is achieved for each of the parameters
$\Delta_{1}$, $\Delta_{2}$, $\Delta_{3}$ and $A$ simultaneously.

After obtaining a self-consistent solution for $\Delta_{\alpha}$ and $A$, the
tunneling conductance in the system may be found as

$\begin{gathered}\frac{\partial I_{i\alpha}\left(V\right)}{\partial
V}\propto\sum_{n}\left[\left|u_{in}\right|^{2}f^{\prime}\left(E_{n}-qV\right)\right.\\\
\left.+\left|v_{in}\right|^{2}f^{\prime}\left(E_{n}+qV\right)\right]\end{gathered}$
(8)

where $f^{\prime}$ is a derivative of the Fermi-function, $i$ indicates the
lattice point, $n$ denotes eigenstate of the system and $V$ is applied voltage
[28, 29, 30].

##  III Results

This paper analyzes the solutions for square systems with linear sizes up to
$L=64$ nodes. We consider system with symmetric intraband interaction
$V_{11}=V_{22}=V_{33}=2.4$ and negative interband coupling
$V_{12}=V_{13}=V_{23}=-0.6$ with fixed dimensionless charge $q=0.25$ and
various temperatures. We also investigate case of slightly non-symmetric
intraband interaction $V_{11}=2.5$, $V_{22}=V_{33}=2.4$,
$V_{12}=V_{13}=V_{23}=-u$ for $u=0.6,\,0.3,\,0.1$. $\epsilon=10^{-8}$,
$\varepsilon=10^{-6}$ are selected as convergence parameters for all
simulations.

We start the iteration procedure with an initial guess for the superconducting
gaps Eq. (5) and vector potential $A$ and converge to a self-consistent
solution. Here, we consider four types of initial conditions, yielding
different objects. To obtain a uniform solution without any vortex or domain
wall, we set $\Delta_{i1}=0.1$, $\Delta_{i2}=0.1\cdot e^{2\pi i/3}$, and
$\Delta_{i3}=0.1\cdot e^{-2\pi i/3}$. A domain wall is generated by the same
initial guess in the upper half of the sample, and with $\Delta_{i2}=0.1\cdot
e^{-2\pi i/3}$, $\Delta_{i3}=0.1\cdot e^{2\pi i/3}$ in the lower one. For
conventional single-flux-quanta vortex each gap was initialized as
$\Delta_{i\alpha}=\tanh\left(20r_{i}/L\right)\cdot
e^{i\varphi_{i}+2\pi\alpha/3}$, where $r_{i}$ and $\varphi_{i}$ are spherical
coordinates with $r=0$ corresponding to the center of the sample and $L$ is
the system size. Unquantized vortex is generated by setting
$\Delta_{i1}=\tanh\left(20r_{i}/L\right)\cdot e^{i\varphi_{i}}$ and two other
bands the same as for the uniform case. The vector potential is set to zero in
the initial conditions in all four cases. Upon convergence, the domain wall is
stable because of geometric pinning. To ensure that the vortex anzats-
initiated configuration converges to a stable vortex solution, we choose a
significantly large grid size relative to solution size so that the
interaction of the vortex with a boundary is smaller than the numerical grid
pinning or numerical accuracy.

###  III.1 Vortex structure

There is a significant difference between the fractional vortex solutions for
systems with (FIG. 2) and without (FIG. 1) Josephson coupling. Let us define
“partial” currents as

$J_{ij\alpha}=-2q\sum_{\sigma}\Im{\expectationvalue{c_{i\sigma\alpha}^{\dagger}c_{j\sigma\alpha}}\exp{\mathrm{i}qA_{ij}}}.$
(9)

In a system where particle transitions between bands ($V_{\alpha,\beta}=0$ if
$\alpha\neq\beta$) are forbidden, intercomponent interaction is induced only
by coupling the vector potential, so the currents are associated with the
independently conserved components for the $U(1)^{3}$ case (FIG. 1). One can
see from the solutions that the component with the phase winding has a
clockwise circulation of current. The other two components have counter-
clockwise circulating currents because they do not have phase winding, and the
current is due only to vector potential (note also the different magnitude of
the current near the vortex core for components without phase winding). That
coexistence of clock- and anti-clock-wise circulation is closely connected
with the fractionalization of flux quantum. Another aspect that can be seen
from the solutions is that the individual currents slowly decrease away from
the vortex core, but the total current is strongly localized. The fact that
far from the fractional vortex core, there are oppositely circulating currents
in different components is the reason why the fractional vortex in this model
has logarithmically divergent energy, as discussed in a different formalism
based on macroscopic London model in [8].

When interband Josephson coupling leads to spontaneous breaking of time-
reversal symmetry, the axial symmetry of partial currents breaks down, shown
on FIG. 2. As will be clear from the images below, oppositely directed partial
currents are associated with the interband phase-difference gradients induced
by the domain walls emitted by a vortex.

Figure 1: Unquantized vortex with phase winding only in the first phase in a
$U(1)^{3}$ three-component model. Current distribution Eq. (6) $J$, partial
currents Eq. (9) $J_{1}$, $J_{2}$, $J_{3}$, absolute gap values $\Delta_{1}$,
$\Delta_{2}$, $\Delta_{3}$, and relative gap phases $\varphi_{1}-\varphi_{2}$
and $\varphi_{1}-\varphi_{3}$ are demonstrated. Square sample with linear size
32 with intraband coupling $V_{11}=V_{22}=V_{33}=2.4$, zero interband
coupling, $T=0.12$ and $q=0.25$ is simulated. Figure 2: Unquantized vortex in
a $U(1)\times Z_{2}$ three-component model. Model parameters are the same, as
FIG. 1, except interband coupling $V_{12}=V_{13}=V_{23}=-0.6$ and temperature
$T=0.36$. On the domain wall, away from the vortex, partial currents
compensate, and the total current is zero

###  III.2 Domain wall

Negative inter-component coupling makes the ground state of the system
frustrated. When all three bands are identical, there are two possible ground
states - with phases of the components
$\phi_{1}-\phi_{2}=\phi_{2}-\phi_{3}=2\pi/3$ and
$\phi_{1}-\phi_{2}=\phi_{2}-\phi_{3}=-2\pi/3$. If both phase-locking states
coexist in the system, a domain wall is formed between the phases.

FIG. 2 shows that on the domain wall, two of the gaps are suppressed, and the
third one is enhanced. So, if the bands are not symmetric, different kinds of
domain walls may emerge. Such domain walls can, in general, have different
energies per unit length. As an example, we calculated domain wall energy
$E=E_{\mathrm{DW}}-E_{\mathrm{U}}$ for several Josephson couplings FIG. 3.
Here $E_{\mathrm{DW}}$ stands for the total free energy Eq. (7) of a system
with a domain wall, and $E_{\mathrm{U}}$ is the total free energy of an empty
system. According to our results, if one of the gaps in the system has a
larger amplitude, the domain wall, on which this gap is enhanced, has lower
energy than a domain wall, where this gap is suppressed. Let us call the first
kind on the domain wall a low energy (LE) domain wall, and the second kind - a
high energy (HE) domain wall.

The fractional vortex may be viewed as a junction point between two different
domain walls. If these walls have different energies per unit length,
effective force, proportional to the energy difference, will be applied to the
vortex. That can make vortices with winding on some bands harder to stabilize
by pinning.

Figure 3: Domain wall energy in bandwidth units as a function of temperature.
Domain wall was generated in the middle of a square sample with $24\times 24$
nodes. Results were obtained for slightly asymmetric intraband coupling
$V_{11}=2.5$, $V_{22}=V_{33}=2.4$ and various interband couplings
$V_{12}=V_{13}=V_{23}=-u$, $u=0.6,\,0.3,\,0.1$. The energy is calculated for
both kinds of domain walls - high energy (HE) and low energy (LE). A
characteristic temperature exists for each coupling strength, above which the
HE domain wall becomes unstable on our numerical grid. For higher
temperatures, domain wall width becomes comparable with the system size.

###  III.3 Tunneling conductance

Scanning Tunneling Microscopy is a powerful technique for studying vortices,
e.g. [31, 32]. In our model, tunneling conductance may be calculated using Eq.
(8). The first question is how observable domain walls in $s+is$
superconductors are via their density-of-states signatures.

FIG. 4 represents tunneling conductance curves for high energy (HE) and low
energy (LE) domain walls, compared with tunneling conductance dependence in
domain-wall-free samples. Domain walls for certain coupling strength $u$ have
noticeable suppression of the gaps and thus may be observed in STM. The domain
wall conductance is higher than the bulk conductance below the superconducting
gap and lower above the gap. It may also be seen that the patterns for both
kinds of domain walls are pretty similar.

Figure 4: Tunneling conductance Eq. (8) for different domain walls and uniform
solution (U). Simulation parameters are the same as on FIG. 3, $T=0.25T_{c}$.
For weak interband coupling $s+is$, the domain wall gives nearly no signature
in tunneling conductance. Figure 5: Tunneling conductance
$\sigma\left(V\right)$ Eq. (8) for conventional $\left(C_{\mathrm{c}}\right)$
and fractional $\left(F_{\mathrm{c}}\right)$ vortex cores and gap amplitude
crossections in $U(1)^{3}$ and $U(1)\times Z_{2}$ systems. A square sample
with a linear size of 64 nodes was analyzed for a system with couplings
$V_{11}=V_{22}=V_{33}=2.4$ and charge $q=0.25$. For $U(1)^{3}$ system
temperature $T=0.37$ and zero intercomponent coupling was used; for
$U(1)\times Z_{2}$ system $T=0.27$ and intercomponent coupling is
$V_{12}=V_{13}=V_{23}=-0.6$.

FIG. 5 demonstartes a significant difference between fractional and
conventional vortex in $U(1)^{3}$ and $U(1)\times Z_{2}$ systems arising due
to very different core structures.

###  III.4 Magnetic field

In general, the shape and localization of the magnetic field of a fractional
vortex in these models depend on the temperature and the strength of inter-
component Josephson coupling.

It was pointed out in the $U(1)\times U(1)$ Ginzburg-Landau models that
general fractional vortices have power-law localization of the magnetic flux
[33]. Also, clear differences in the magnetic field location of fractional
versus integer vortices were observed in the experiment [9]. In the case of
the Josephson-coupled system, the phase-difference mode is massive. Hence, we
expect an exponential localization, but the overall localization can differ.

Concerning the shape of the magnetic field, the following patterns can be
observed - conventional vortex in both interacting and non-interacting systems
is rotationally symmetric (apart from some corrections related to the square
symmetry of our grid) and carries one quantum of magnetic flux FIG. 6, 7. The
unquantized vortex carries $1/3$ quanta in both cases due to the chosen
symmetry between the components. However, in the presence of Josephson
coupling, it generally tends to elongate along the domain wall.

Figure 6: Magnetic field for conventional (CV) and fractional (FV) vortices in
$U(1)^{3}$ system. Simulation parameters are the same as on FIG. 5. Both
vortices have rotational symmetry. Figure 7: Magnetic field for conventional
(CV) and fractional (FV) vortices in $U(1)\times Z_{2}$ system. Note that
despite the difference in the field amplitude, the localization of the
magnetic field of the fractional vortex is different. The model parameters are
the same as on FIG. 5. The fractional vortex is elongated along the domain
wall.

##  IV Discussion

The observation of vortices that carry a varying fraction of quantum flux
(unquantized vortices) in the multiband superconductor Ba1-xKxFe2As2 requires
a theoretical understanding of these objects. The non-trivial aspect of this
material is the existence of only two broken symmetries but at least three
superconducting bands. According to the Ginzburg-Landau and London models-
based arguments [8, 22], this circumstance may lead to more than two
fractional vortices. One cannot justify the existence of these objects by
standard symmetry- and topology-based arguments; hence, microscopic
justification is especially important. This paper studied the solutions for
fractional vortices and domain walls in a fully microscopic three-band
Bogoliubov-de-Gennes model, including a fully self-consistent solution for
magnetic fields. We find stable vortices whose magnetic field shape,
localization, and density-of-states signatures significantly differ from the
single-quanta vortices and can be directly probed in experiments. In the model
of $s+is$ superconductors, we find that domain walls give a signature in
density-of-states. However, this signature is parameters-dependent and may be
undetectable. The experimental characterization of these vortices is important
also because enclosing a variable fraction of the flux quantum makes them
realize the charge-flux-tube bound state [11]. Hence, these objects obey
fractional statistics. Another application for these objects is fluxonics [34,
35].

###### Acknowledgements.

We thank A. Benfenati and M. Barkman for insightful discussions and providing
the code developed in [9]. This work was supported by the Swedish Research
Council Grants 2022-04763, by Olle Engkvists Stiftelse, and the Wallenberg
Initiative Materials Science for Sustainability (WISE) funded by the Knut and
Alice Wallenberg Foundation.

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