** with Alberto Corticelli, Mihail Silaev and Egor Babaev **

Disorder in two-band superconductors with repulsive interband interaction
induces a frustrated competition between the phase-locking preferences of
the various potential and kinetic terms. This frustrated interaction can
result in the formation of an $s+is$ superconducting state, that breaks the
time-reversal symmetry.
In this paper we study the normal modes and their associated coherence lengths
in such materials. We especially focus on the consequences of the soft modes
stemming from the frustration and time-reversal-symmetry breakdown. We find that
two-bands superconductors with such impurity-induced frustrated interactions
display a rich spectrum of physical properties that are absent in their clean
counterparts.
It features a mixing of Leggett's and Anderson-Higgs modes, and a soft mode
with diverging coherence length at the impurity-induced second order phase
transition to the $s+is$ state. Such a soft mode generically results in
long-range attractive intervortex forces that can trigger the formation
of vortex clusters. We find that, if such
clusters are formed, their size and internal flux density have a
characteristic temperature dependence that could be probed in
muon-spin-rotation experiments.

Preprint on
arXiv (2018)

** with Alberto Corticelli, Mihail Silaev and Egor Babaev **

In multiband systems, such as iron-based superconductors, the superconducting
states with locking and anti-locking of the interband phase differences,
are usually considered as mutually exclusive. For example, a dirty two-band
system with interband impurity scattering undergoes a sharp crossover between
the $s_{\pm}$ state (that favors phase anti-locking) and the $s_{++}$ state
(that favors phase locking).
We discuss here that the situation can be much more complex in the presence
of an external field or superconducting currents. In an external applied
magnetic field, dirty two-band superconductors do not feature a sharp
$s_{\pm}\to s_{++}$ crossover but rather a washed-out crossover to a finite
region in the parameter space where both $s_{\pm}$ and $s_{++}$ states can
coexist for example as a lattice or a microemulsion of inclusions of different
states.
The current-carrying regions such as the regions near vortex cores can exhibit
$s_\pm$ state while it is the $s_{++}$ state that is favored in the bulk.
This coexistence of both states can even be realized in the Meissner state
at the sample's boundaries featuring Meissner currents. We demonstrate that
there is a magnetic-field-driven crossover between the pure $s_{\pm}$ and the
$s_{++}$ states.

Preprint on
arXiv (2017)

** with A. A. Zyuzin and Egor Babaev **

We study topological excitations in two-component nematic superconductors,
with a particular focus on Cu$_x$Bi$_2$Se$_3$ as a candidate material.
We find that the lowest-energy topological excitations are coreless vortices:
a bound state of two spatially separated half-quantum vortices. These objects
are nematic Skyrmions, since they are characterized by an additional
topological charge.
The inter-Skyrmion forces are dipolar in this model, i.e. attractive for
certain relative orientations of the Skyrmions, hence forming multi-Skyrmion
bound states.

Published in
Phys. Rev. Lett. **96**, 140503 (2017)

** with Mihail Silaev and Egor Babaev **

We report a nontrivial transition in the core structure of vortices in
two-band superconductors as a function of interband impurity scattering.
We demonstrate that, in addition to singular zeros of the order parameter,
the vortices there can acquire a circular nodal line around the singular
point in one of the superconducting components. It results in the formation
of the peculiar ``moat"-like profile in one of the superconducting gaps.
The moat-core vortices occur generically in the vicinity of the
impurity-induced crossover between $s_{\pm}$ and $s_{++}$ states.

Published in
Phys. Rev. B **96**, 140503(R) (2017)

** with Mihail Silaev and Egor Babaev **

We investigate the phase diagram of dirty two-band superconductors.
This paper primarily focuses on the properties and observability of the
time-reversal symmetry-breaking $s+is$ superconducting states, which
can be generated in two-band superconductors by interband impurity
scattering. We show that such states can appear in two distinct ways.
First, according to a previously discussed scenario, the $s+is$ state can
form as an intermediate phase at the impurity-driven crossover between
$s_{\pm}$ and $s_{++}$ states. We show that there is a second scenario
where domains of the $s+is$ state exists in the form of an isolated
dome inside the $s_{\pm}$ domain, completely detached from the transition
between $s_{\pm}$ and $s_{++}$ states. We demonstrate that in both cases
the $s+is$ state, generated by impurity scattering exists in an extremely
small interval of impurity concentrations. Although this likely precludes
direct experimental observation of the $s+is$ state formation due to this
mechanism, this physics leads to the appearance of a region inside both
the $s_{\pm}$ and $s_{++}$ domains with unusual properties due to
softening of normal modes.

Published in
Phys. Rev. B **95**, 024517 (2017)

** with Egor Babaev, Troels Arnfred Bojesen and Asle Sudbø**

We investigate the magnetization processes of a standard Ginzburg-Landau model for
chiral $p$-wave superconducting states in an applied magnetic field. We find that
the phase diagram is dominated by triangular lattices of doubly quantized vortices.
Only in close vicinity to the upper critical field, the lattice starts to dissociate
into a structure of single-quanta vortices.
The degeneracy between states with opposite chirality is broken in a nonzero field.
If the magnetization starts with an energetically unfavorable chirality, the process
of chirality-inversion induced by the external magnetic field results in the formation
of a sequence of metastable states with characteristic magnetic signatures that can
be probed by standard experimental techniques.

Published in
Phys. Rev. B **94**, 104509 (2016)

** with Mihail Silaev and Egor Babaev **

Starting with the generic Ginzburg-Landau expansion from a microscopic $N$-band
model, we focus on the case of a 3-band model which was suggested to be relevant
to describe some iron-based superconductors. This can lead to the so-called $s+is$
superconducting state that breaks time-reversal symmetry due to the competition
between different pairing channels. Of particular interest in that context, is
the case of an interband dominated pairing with repulsion between different bands.
For that case we consider in detail the relevant reduced two-component Ginzburg-Landau
theory. We provide detailed analysis of the ground state, length scales and topological
properties of that model.

* Prepared for the proceedings of Vortex IX conference in Rhodes (Sept. 2015). *

Published in
Physica C: Superconductivity and its Applications, **533**, 63-73 (2017)

** with Mihail Silaev and Egor Babaev **

We show that superconductors with broken time-reversal symmetry have very specific
magnetic and electric responses to inhomogeneous heating. A local heating of such
superconductors induces a magnetic field with a profile that is sensitive to the
presence of domain walls and crystalline anisotropy of superconducting states.
A nonstationary heating process produces an electric field and a charge imbalance
in different bands. These effects can be measured and used to distinguish $s+is$
and $s+id$ superconducting states in the candidate materials such as
Ba$_{1-x}$K$_x$Fe$_2$As$_2$.

Published in
Phys. Rev. Lett. **116**, 097002 (2016)

** with Egor Babaev **

Chiral $p$-wave superconducting state supports a rich spectrum of
topological excitations different from those in conventional
superconducting states. Besides domain walls separating different
chiral states, chiral $p$-wave state supports both singular and
coreless vortices also interpreted as skyrmions.
Here, we present a numerical study of the energetic properties of
isolated singular and coreless vortex states as functions of
anisotropy and magnetic field penetration length. In a given chiral
state, single quantum vortices with opposite winding have different
energies and thus only one kind is energetically favoured. We find
that with the appropriate sign of the phase winding, two-quanta
(coreless) vortices are always energetically preferred over two isolated
single quanta (singular) vortices. We also report solutions carrying
more flux quanta. However those are typically more energetically
expensive/metastable as compared to those carrying two flux quanta.

Published in
Scientific Reports **5**, 17540 (2015)

** with Mihail Silaev and Egor Babaev **

We demonstrate that superconductors which break time-reversal
symmetry can exhibit thermoelectric properties, which are entirely
different from the Ginzburg mechanism. As an example, we show
that in the $s+is$ superconducting state there is a reversible
contribution to thermally induced supercurrent, whose direction
is not invariant under time-reversal operation. Moreover in
contrast to Ginzburg's mechanism it has a singular behavior near
the time-reversal symmetry breaking phase transition. The effect
can be used to confirm or rule out the $s+is$ state, which is
widely expected to be realized in pnictide compounds
Ba$_{1-x}$K$_{x}$Fe$_2$As$_2$ and stoichiometric LiFeAs.

Published in
Phys. Rev. B **91**, 104512 (2015)

** with Daniel F. Agterberg **

We consider competing pair-density-wave (PDW) and d-wave superconducting
states in a magnetic field. We show that PDW order appears in the cores of
$d$-wave vortices, driving checkerboard charge-density-wave (CDW) order in
the vortex cores, which is consistent with experimental observations.
Furthermore, we find an additional CDW order that appears on a ring outside
the vortex cores. This CDW order varies with a period that is twice that of
the checkerboard CDW and it only appears where both PDW and $d$-wave order
coexist. The observation of this additional CDW order would provide strong
evidence for PDW order in the pseudogap phase of the cuprates. We further
argue that the CDW seen by nuclear magnetic resonance at high fields is due
to a PDW state that emerges when a magnetic field is applied.

Published in
Phys. Rev. B **91**, 104512 (2015)

** with Egor Babaev **

We study superconductors with two order components and phase separation
driven by intercomponent density-density interaction, focusing on the phase
where only one condensate has nonzero ground-state density and a competing
order parameter exists only in vortex cores.
We demonstrate there that multibody intervortex interactions can be
strongly nonpairwise, leading to some unusual vortex patterns in an
external field, such as vortex pairs and vortex chains.
We demonstrate that in an external magnetic field such a system undergoes
a field-driven phase transition from (broken) $U(1)$ to (broken) $U(1)\times U(1)$
symmetries when a subdominant order parameter in the vortex cores acquires
global coherence. Observation of these characteristic ordering patterns
in surface probes may signal the presence of a subdominant condensate
in the vortex core.

Published in
Phys. Rev. B **91**, 014510 (2014)

** with Egor Babaev **

We study the properties of vortex solutions and magnetic response
of two-component $\mathrm{U}(1)\times \mathrm{U}(1)\times\mathbb{Z}_2$
superconductors, with phase separation driven by intercomponent
density-density interaction. Such a theory can be viewed arising
from the breakdown of $SU(2)$ symmetry by a biquadratic
interaction between the components of the field. Depending on the
symmetrry-breaking term, there are two ground-state phases: one where
both components of the doublet are equal (the miscible phase) and one
where only one component assumes a non zero vacuum expectation value
(the immiscible state). In the latter phase, the spectrum of topological
excitations contains both domain walls and vortices.
We show the existence of another kind of excitation that has properties
of both topological excitations at the same time. They combine
vorticity together with a circular domain wall, interpolating
between inequivalent broken states, that shows up as a ring of localized
magnetic flux.
Asymptotically, this resembles a vortex carrying multiple flux quanta,
but because the magnetic field is localized at a given distance from
the center this looks like a pipe. The isolated multiquanta pipelike
vortices can be either stable or metastable, even if the system is not
type-1. We also discuss the response of such a system to an externally
applied magnetic field.

Published in
Phys. Rev. B **90**, 214524 (2014)

** with Daniel Agterberg and Egor Babaev **

When an in-plane field is applied to a clean interface superconductor,
a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like phase is stabilized.
This phase has a $ \mathrm{U}(1)\times\mathrm{U}(1) $ symmetry and, in
principle, this symmetry allows for flux carrying topological excitations
different from Abrikosov vortices (which are the simplest defects associated
with $ S^1\to S^1 $ maps). However, in practice, largely due to electromagnetic
and other intercomponent interactions, such topological excitations are very
rare in superconducting systems. Here we demonstrate that a realistic
microscopic theory for interface superconductors, such as SrTiO$_3$/LaAlO$_3$ ,
predicts an unconventional magnetic response where the flux-carrying objects
are skyrmions, characterized by homotopy invariants of $ S^2\to S^2 $ maps.
Additionally, we show that this microscopic theory predicts that stable fractional
vortices form near the boundary of these superconductors. It also predicts
the appearance of type-1.5 superconductivity for some range of parameters.
Central to these results is the assumption that the Rashba spin orbit coupling
is much larger than the superconducting gap.

Published in
Phys. Rev. B **90**, 064509 (2014)

** with Egor Babaev**

We investigate the topological defects in phenomenological models describing
mixtures of charged condensates with commensurate electric charges. Such situations
are expected to appear for example in liquid metallic deuterium. This is modeled by
a multi-component Ginzburg-Landau theory where the condensates are coupled to the
same gauge field by different coupling constants whose ratio is a rational number.
We also briefly discuss the case where electric charges are incommensurate. Flux
quantization and finiteness of the energy per unit length dictates that the different
condensates have different winding and thus different number of (fractional) vortices.
Competing attractive and repulsive interactions lead to molecule-like bound state between
fractional vortices. Such bound states have finite energy and carry integer flux quantum.
These can be characterized by $\mathbb{C}P^1$ topological invariant that motivates their
denomination as skyrmions.

Published in
Phys. Rev. B **89**, 214507 (2014)

** with Egor Babaev**

Arguments were recently advanced that hole-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$
exhibits the $s+is$ state at certain doping. Spontaneous breaking of
time-reversal symmetry in the $s+is$ state dictates that it possess domain
wall excitations. Here, we discuss what are the experimentally detectable
signatures of domain walls in the $s+is$ state. We find that in this state
the domain walls can have a dipolelike magnetic signature (in contrast to
the uniform magnetic signature of domain walls $p+ip$ superconductors).
We propose experiments where quench-induced domain walls can be stabilized
by geometric barriers and observed via their magnetic signature or their
influence on the magnetization process, thereby providing an experimental
tool to confirm the $s+is$ state.

Published in
Phys. Rev. Lett. **112**, 017003 (2014)

** with Karl Sellin, Juha Jäykkä and Egor Babaev**

Rather generically, multicomponent superconductors and superfluids have intercomponent
current-current interaction. This applies to wide range of physical systems from mixtures
of hadronic superfluids in neutron stars, ultracold atoms to $p$-wave superconductors.
Here we discuss how this kind of interaction affects magnetic response of superconductors.
Going beyond the frequently used London limit, we show that in superconductors with
substantially strong intercomponent drag interaction, the topological defects which form
in external field are Skyrmions. We study their properties and show that they can be
distinguished from ordinary vortices by a very characteristic magnetization process.

Published in
Phys. Rev. B **89**, 104508 (2014)

** with Eugen Radu and Mikhail S. Volkov**

We present for the first time solutions in the gauged $U(1)\times U(1)$ model of Witten
describing vortons -- spinning flux loops stabilized against contraction by the
centrifugal force. Vortons were heuristically described many years ago, however,
the corresponding field theory solutions were not obtained and so the stability
issue remained open. We construct explicitly a family of stationary vortons
characterized by their charge and angular momentum. Most of them are unstable and
break in pieces when perturbed. However, thick vortons with small radius preserve
their form in the $3+1$ non-linear dynamical evolution. This gives the first ever
evidence of stable vortons and impacts several branches of physics where they
could potentially exist. These range from cosmology, where vortons could perhaps
account for the cold dark matter, to condensed matter physics.

Published in
Phys. Rev. Lett. **111**, 171602 (2013)

** with Johan Carlström, Egor Babaev and Martin Speight**

It is shown that under certain conditions, three-component superconductors
(and in particular three-band systems) allow stable topological defects different
from vortices. We demonstrate the existence of these excitations, characterized by a
$\ {\mathbb{C}}{{P}}^2\ $ topological invariant, in models for three-component
superconductors with broken time reversal symmetry. We term these topological defects
"chiral $GL^{(3)}$ skyrmions", where "chiral" refers to the fact that due to broken
time reversal symmetry, these defects come in inequivalent left- and right-handed versions.
In certain cases these objects are energetically cheaper than vortices and should be
induced by an applied magnetic field. In other situations these skyrmions are metastable
states, which can be produced by a quench. Observation of these defects can signal
broken time reversal symmetry in three-band superconductors or in Josephson-coupled
bilayers of $s_\pm$ and $s$-wave superconductors.

Published in
Phys. Rev. B **87**, 014507 (2013)

** with Daniel Agterberg and Egor Babaev **

Recently vortex coalescence was reported in superconducting
Sr$_2$RuO$_4$ by several experimental groups for fields applied along
the $c$ axis. We argue that Sr$_2$RuO$_4$ is a type-1.5 superconductor
with long-range attractive, short-range repulsive intervortex interaction.
The type-1.5 behavior stems from an interplay of the two orbital degrees
of freedom describing this chiral superconductor together with the multiband
nature of the superconductivity. These multiple degrees of freedom give
rise to multiple coherence lengths, some larger and some smaller than
the magnetic field penetration length, resulting in nonmonotonic
intervortex forces.

Published in
Phys. Rev. B **86**, 060513(R) (2012)

Article relegated in Russian popular science article Link

** with Egor Babaev **

Observability of half-quantum vortices and skyrmions in $p$-wave
superconductors is an outstanding open question. Under the most common
conditions, fractional flux vortices are not thermodynamically stable
in bulk samples. Here we show that in chiral $p$-wave superconductors,
there is a regime where, in contrast, lattices of integer-flux vortices
are not thermodynamically stable. Instead, skyrmions made of spatially
separated half-quantum vortices are the topological defects produced by
an applied external field.

Published in
Phys. Rev. B **86**, 060514(R) (2012)

** with Johan Carlström and Egor Babaev **

A popular article on our research on type-1.5 superconductivity

**(PhysOrg.com) --** “*In this 100th anniversary year of the discovery
of superconductivity, physicists at the University of Massachusetts
Amherst and Swedenâ€™s Royal Institute of Technology have published a
fully self-consistent theory of the new kind of superconducting
behavior, Type 1.5, this month in the journal Physical Review B.*”

Read more on
phys.org

Article relegated in several popular science media : cryogenicsociety.org , rdmag.com , physicsforme , newswise.com , zeitnews.org , lescienze.it , engineersonline.nl , sverigesradio.se , mk.ru , sci.ipv2.info , donmarket.org , physics.com.ua , pavlonews.info , horoshienovosti.com.ua , scirt.ru , physiclib.ru , rsci.ru , nanonewsnet.ru , naukoved.ru , membrana.ru

** with E. Babaev , J. Carlström, M. Silaev and J. M. Speight **

A conventional superconductor is described by a single complex order parameter field
which has two fundamental length scales, the magnetic field penetration depth $\lambda$
and the coherence length $\xi$. Their ratio $\kappa$ determines the response of a superconductor
to an external field, sorting them into two categories as follows; type-I when $\kappa < 1 / \sqrt{2}$
and type-II when $\kappa > 1/\sqrt{2}$. We overview here multicomponent systems which can possess three
or more fundamental length scales and allow a separate &ldquo type-1.5 &ldquo superconducting state when, e.g.
in two-component case $\xi_1<\sqrt{2}\lambda< \xi_2$.
In that state, as a consequence of the extra fundamental
length scale, vortices attract one another at long range but repel at shorter ranges. As a consequence the system
should form an additional Semi-Meissner state which properties we discuss below. In that state vortices form clusters
in low magnetic fields. Inside the cluster one of the component is depleted and the superconductor-to-normal interface
has negative energy. In contrast the current in second component is mostly concentrated on the cluster's boundary,
making the energy of this interface positive. Here we briefly overview recent developments in Ginzburg-Landau and
microscopic descriptions of this state.

Published in
Physica C **479**, 2-14 (2012)

** with Johan Carlström and Egor Babaev **

We show that three-band superconductors with broken time reversal symmetry allow magnetic
flux-carrying stable topological solitons which have only a slightly higher energy than ordinary vortices.
Therefore they can be induced by fluctuations or quenching the system through a phase transition. It can
provide an experimental signature of the time reversal symmetry breakdown.

Published in
Phys. Rev. Lett. **107**, 197001 (2011)

** with Johan Carlström and Egor Babaev **

The recent discovery of iron pnictide superconductors has resulted in a rapidly growing
interest in multiband models with more than two bands. In this work we specifically focus on the
properties of three-band Ginzburg-Landau models which do not have direct counterparts in more studied
two-band models. First we derive normal modes and characteristic length scales in the conventional $U(1)$
three-band Ginzburg-Landau model as well as in its time reversal symmetry broken counterpart with
$U(1)\times Z_2$ symmetry. We show that in the latter case, the normal modes are mixed phase/density
collective excitations. A possibility of the appearance of a massless phase-difference mode associated
with fluctuations of the phase difference is also discussed. Next we show that gradients of densities
and phase differences can be inextricably intertwined in vortex excitations in three-band models. This
can lead to very long-range attractive intervortex interactions and appearance of type-1.5 regimes even
when the intercomponent Josephson coupling is large. In some cases it also results in the formation of
a domain-like structures in the form of a ring of suppressed density around a vortex across which one
of the phases shifts by $\pi$. We also show that field-induced vortices can lead to a change of broken
symmetry from $U(1)$ to $U(1)\times Z_2$ in the system. In the type-1.5 regime, it results in a semi-Meissner
state where the system has a macroscopic phase separation in domains with broken $U(1)$ and $U(1)\times Z_2$ symmetries.

Published in
Phys. Rev. B **84**, 134518 (2011)

** with Johan Carlström and Egor Babaev **

We demonstrate existence of non-pairwise interaction forces between vortices in
multicomponent and layered superconducting systems. That is, in contrast to most
common models, the interactions in a group of such vortices is not a universal
superposition of Coulomb or Yukawa forces. Next we consider the properties of vortex
clusters in Semi-Meissner state of type-1.5 two-component superconductors. We show that
under certain condition non-pairwise forces can contribute to formation of very complex
vortex states in type-1.5 regimes.

Published in
Phys. Rev. B **84**, 134515 (2011)