# In Print

## Superconducting phases and the second Josephson harmonic in tunnel junctions between diffusive superconductors

18 June in 12:30

__A.S. Osin__, Ya.V. Fominov

We consider a planar SIS-type Josephson junction between diffusive superconductors (S) through an insulating tunnel interface (I). We construct fully self-consistent perturbation theory with respect to the interface conductance. As a result, we find correction to the first Josephson harmonic and calculate the second Josephson harmonic. At arbitrary temperatures, we correct previous results for the nonsinusoidal current-phase relation in Josephson tunnel junctions, which were obtained with the help of conjectured form of solution. Our perturbation theory also describes the difference between the phases of the order parameter and of the anomalous Green functions. The talk is based on the paper [1].

[1] A.S. Osin and Ya.V. Fominov, arXiv:2105.05786

[1] A.S. Osin and Ya.V. Fominov, arXiv:2105.05786

## Relation between multifractality and entanglement for nonergodic extended states

18 June in 11:30

Ivan M. Khaymovich (MPIKS, Dresden)

The multifractality provides a way of ergodicity breaking in term of chaotization and equipartitioning over degrees of freedom. On the other hand, in quantum information theory it is the entanglement entropy which represents the main measure of ergodicity and thermalization.
In this talk I will represent an exact relation between the above measures, showing that the fractal dimension of the non-ergodic wave function puts an upper bound on its entanglement entropy [A]. I will also provide a couple of explicit examples demonstrating that the entanglement entropy may reach its ergodic (Page) value when the wave function is still highly non-ergodic and occupies a zero fraction of the total Hilbert space.
If time permits I will briefly discuss some other possible deviations from ergodicity relevant for the chaotic many-body systems [B-E].

[A] G. De Tomasi, I. M. K., “Multifractality meets entanglement: relation for non-ergodic extended states”, Phys. Rev. Lett. 124, 200602 (2020) [arXiv:2001.03173]

[B] I. M. K., M. Haque, and P. McClarty, “Eigenstate Thermalization, Random Matrix Theory and Behemoths”, Phys. Rev. Lett. 122, 070601 (2019) [arXiv:1806.09631].

[C] M. Haque, P. A. McClarty, I. M. K. , “Entanglement of mid-spectrum eigenstates of chaotic many-body systems—deviation from random ensembles.” [arXiv:2008.12782].

[D] A. Bäcker, I. M. K., M. Haque,, “Multifractal dimensions for chaotic quantum maps and many-body systems”, Phys. Rev. E 100, 032117 (2019) [arxiv:1905.03099].

[E] G. De Tomasi, I. M. K. , "Ergodic Entanglement of many-body multifractal states in quadratic Hamiltonians", in preparation

[A] G. De Tomasi, I. M. K., “Multifractality meets entanglement: relation for non-ergodic extended states”, Phys. Rev. Lett. 124, 200602 (2020) [arXiv:2001.03173]

[B] I. M. K., M. Haque, and P. McClarty, “Eigenstate Thermalization, Random Matrix Theory and Behemoths”, Phys. Rev. Lett. 122, 070601 (2019) [arXiv:1806.09631].

[C] M. Haque, P. A. McClarty, I. M. K. , “Entanglement of mid-spectrum eigenstates of chaotic many-body systems—deviation from random ensembles.” [arXiv:2008.12782].

[D] A. Bäcker, I. M. K., M. Haque,, “Multifractal dimensions for chaotic quantum maps and many-body systems”, Phys. Rev. E 100, 032117 (2019) [arxiv:1905.03099].

[E] G. De Tomasi, I. M. K. , "Ergodic Entanglement of many-body multifractal states in quadratic Hamiltonians", in preparation

## Interaction of a Neel-type skyrmion and a superconducting vortex

11 June in 11:30

__E.S. Andriyahina__, I.S. Burmistrov

Superconductor-ferromagnet heterostructures hosting vortices and skyrmions are new area of an interplay between superconductivity and magnetism. We study [1] an interaction of a Neel-type skyrmion and a Pearl vortex in thin heterostructures due to stray fields. Surprisingly, we find that it can be energetically favorable for the Pearl vortex to be situated at some nonzero distance from the center of the Neel-type skyrmion. The presence of a vortex-antivortex pair is found to result in increase of the skyrmion radius. Our theory predicts that a spontaneous generation of a vortex-anti-vortex pair is possible under some conditions in the presence of a Neel-type skyrmion that is in agreement with recent experiment [2].

[1] E.S. Andriyakhina. and I.S. Burmistrov “Interaction of a Néel–type skyrmion and a superconducting vortex” // arXiv: https://arxiv.org/abs/2102.05434.

[2] A.P. Petrović et al. “Skyrmion-(Anti)vortex coupling in a chiral magnet-superconductor heterostructure” // Phys.Rev. Lett.126, 117205 (2021).

[1] E.S. Andriyakhina. and I.S. Burmistrov “Interaction of a Néel–type skyrmion and a superconducting vortex” // arXiv: https://arxiv.org/abs/2102.05434.

[2] A.P. Petrović et al. “Skyrmion-(Anti)vortex coupling in a chiral magnet-superconductor heterostructure” // Phys.Rev. Lett.126, 117205 (2021).

## Vortex structure in a clean superconductor in the vicinity of a planar defect

11 June in 11:30

__U.E. Khodaeva__, M.A. Skvortsov

We investigate the structure of the quasiparticle states localized in a
superconducting vortex core in the vicinity of a planar defect. It is
shown that even a highly transparent defect leads to a significant
modification of the excitation spectrum, with the opening of a minigap
at the Fermi energy. The magnitude of the minigap exceeds the mean level
spacing for a clean vortex already for small values of the reflection
coefficient off the defect. It is maximal for the vortex sitting right
at the defect, decreases with increasing the distance from the defect
and closes at some point. We then generalize the problem for various
configurations of several linear defects (periodic structures, two
crossing lines, stars). Though the minigap remains, a strong
commensurability effect is observed. For two crossing linear defects,
the magnitude of the minigap strongly depends on how close the
intersection angle is to a rational number.

## Dynamical phases in Rosenzweig-Porter model with a ’multifractal’ distribution of hopping

4 June in 11:30

Vladimir Kravtsov

We consider a Rosenzweig-Porter (RP) random matrix model with broad
distribution of off-diagonal matrix elements which emerge as a model
equivalent to the Anderson model on random regular graph. In this work
we study the survival probability of a quantum particle with wave
function initially localized on one site described by such type of model
and show that the strongly non-Gaussian, broad distribution of hopping
leads to the stretch-exponential decay of survival probability with
time. We show that the ergodic phase with the stretch-exponential
behavior of survival probability emerges in this model exactly where on
a finite Bethe lattice there is a multifractal phase and relate the
stretch exponent $\kappa$ with the kernel \epsilon_{\beta} of the
transfer-matrix equation on a Bethe lattice.

We also consider the extension of this RP model relaxing the symmetry inherited from the Bethe lattice and find lines of phase transitions from the exponential to the stretch exponential behavior of survival probability.

We also consider the extension of this RP model relaxing the symmetry inherited from the Bethe lattice and find lines of phase transitions from the exponential to the stretch exponential behavior of survival probability.

## Microwave response of a chiral Majorana interferometer

14 May in 11:30

Alexander Shnirman (Karlsruhe Institute of Technology)

We consider an interferometer based on artificially induced topological superconductivity and chiral 1D Majorana fermions. The (non-topological) superconducting island inducing the superconducting correlations in the topological substrate is assumed to be floating. This allows probing the physics of interfering Majorana modes via microwave response, i.e., the frequency dependent impedance between the island and the earth. Namely, charging and discharging of the island is controlled by the time-delayed interference of chiral Majorana excitations in both normal and Andreev channels. We argue that microwave measurements provide a direct way to observe the physics of 1D chiral Majorana modes.

## Two-impurity scattering in quasi-one-dimensional systems

14 May in 11:30

A.S. Ioselevich,

__N.S. Peshcherenko__
In a quasi-one-dimensional system (a tube) with low concentration of defects $n$ the resistivity $\rho$ has peaks (van-Hove singularities) as a function of Fermi-energy. We show that due to non-Born scattering effects a deep narrow gap should appear just in the center of each peak. The resistivity at the bottom of a gap ($\rho_{\min}\propto n^2$) is dominated by scattering at rare ``twin-pairs'' of close defects, while scattering at solitary defects is suppressed. This effect is characteristic for multi-channel systems, it can not be observed in strictly one-dimensional one.

## Conductivity and thermoelectric coefficients of doped SrTiO_{3} at high temperatures

30 April in 11:30

__Kh. Nazaryan, M.V. Feigelman__

We developed a theory of electric and thermoelectric conductivity of lightly doped SrTiO3 in the non-degenerate region k_B T ≥ E_F , assuming that the major source of electron scattering is their interaction with soft transverse optical phonons present due to proximity to ferroelectric transition. We have used kinetic equation approach within relation-time approximation and we have determined energy-dependent transport relaxation time τ (E) by the iterative procedure. Using electron effective
mass m and electron-transverse phonon coupling constant λ as two fitting parameters, we are able to describe quantitatively a large set of the measured temperature dependences of resistivity R(T ) and Seebeck coefficient S(T ) for a broad range of electron densities studied experimentally in recent paper by K.Behnia and his colleagues. In addition, we calculated Nernst ratio in the linear approximation over weak magnetic field in the same temperature range.

## Multiple Equilibria and Resilience in Large Complex Systems: beyond May-Wigner model

9 April in 11:30

Yan Fyodorov

We consider two different models of randomly coupled N>>1 autonomous differential equationswith the aim of counting fixed points (aka equilibria), and classifying them by their ''instability index'', i.e. the number of unstable directions.
In the first model, characterized by both translational and rotational statistical symmetry of the vector field, we estimate the probability of an equilibrium to have a given index in a phase with exponentially many equilibria.
In the second model, characterized by only rotational statistical symmetry around a chosen stable equilibrium, we find a characteristic distance beyond which the multitude of equilibria prevents a trajectory to go towards the stable equilibrium. This may shed light on ''resilience'' mechanisms of complex ecosystems.

## Six-waves kinetic equation and its applicability to quintic NLSE

9 April in 11:30 (short)

J. Banks, T. Buckmaster,

__A.O. Korotkevich__, G. Kovacic, J. Shatah
We consider kinetic equation for quintic-NLSE, corresponding to 6-wave nonlinear waves interaction. Kinetic equation for the periodic boundary conditions was derived. We propose conditions on equation parameters which guarantee reasonably accurate description of the wave field by wave kinetic equation. Direct numerical simulations in both dynamical equation and wave kinetic equation frameworks demonstrate acceptable correspondence between these significantly different models when proposed conditions for parameters are satisfied.

## On coincidence of periods of the Berglund–Hübsch–Krawitz (BHK) multiple mirrors

2 April in 11:30 (short)

A.A. Belavin

We consider the multiple Calaby-Yau (CY) mirror phenomenon which appears in Berglund-Hübsch-Krawitz (BHK) mirror symmetry. We show that the periods of the holomor phic nonvanishing form of different Calabi-Yau orbifolds, which are BHK mirrors of the the same CY family, coincide.

Alexander Belavin, Vladimir Belavin, Gleb Koshevoy, "Periods of Berglund–Hübsch–Krawitz mirrors", arXiv: hep-th-2012.03320 ; "Periods of the multiple Berglund–Hübsch–Krawitz mirrors", Letters in Mathematical Physics, 111, 93 (2021).

Alexander Belavin, Vladimir Belavin, Gleb Koshevoy, "Periods of Berglund–Hübsch–Krawitz mirrors", arXiv: hep-th-2012.03320 ; "Periods of the multiple Berglund–Hübsch–Krawitz mirrors", Letters in Mathematical Physics, 111, 93 (2021).