# Seminars at the Landau Institute scientific council

Seminars are held on Fridays in the conference hall of Landau Institute for Theoretical Physics in Chernogolovka, beginning at 11:30.

## Bogoyavlensky lattices and the generalized Catalan numbers

2 December, today in 11:30 (short)

Several years ago A.B. Shabat proposed the problem on the decay of the unit step solution for the Volterra lattice terminated on a half-line. It is resembling the Gurevich-Pitaevsky problem on the step-like solutions for the KdV equation, but it turned out to be simpler since the answer is found explicitly. One solution method is based on the observation that the Taylor series for the tau function of the lattice equation serves as the exponential generating function for the Catalan numbers and is expressed in terms of a hypergeometric function. This can be proved using the well-known result in combinatorics that the Hankel transform for the Catalan numbers is the identity sequence. The second method uses a finite-dimensional reduction associated with the master-symmetry of the lattice; the solution with the unit step initial data is contained within this reduction. This talk is about similar results on the relations between the Bogoyavlensky lattices, the generalized Catalan numbers (known in combinatorics) and the generalized hypergeometric functions.

## Semiclassical approach to calculation of form factors in the sinh-Gordon model

2 December, today in 11:30

M. Lashkevich, O. Lisovyy, T. Ushakova

Form factors in the sinh-Gordon model are studied semiclassically for small values of the parameter $b$ on the background of a radial symmetric classical solution. For this purpose we use a generalization of the radial quantization, well known for a massless boson field. We obtain new special functions, which generalize the Bessel functions and have an interesting interpretation in the theory of the classical sinh-Gordon model. The form factors of the exponential operators are completely determined by classical solutions in the leading order, while the form factors of descendant operators contain quantum corrections even in the leading order. Consideration of descendant operators in two chiralities demands renormalizations, which are analogous to those in the conformal perturbation theory.

## Understanding quantum and classical chaos in Hamiltonian systems through adiabatic transformations

2 December, today in 16:00

A. Polkovnikov (Boston University, USA)

Chaos is synonymous to unpredictability. In the case of classical systems this unpredictability is expressed through exponential sensitivity of trajectories to tiny fluctuations of the Hamiltonian or to the initial conditions. It is well known that chaos is closely related to ergodicity or emergence of statistical mechanics at long times, but the precise relations between them are still debated. In quantum systems the situation is even more controversial with trajectories being ill-defined. A standard approach to defining quantum chaos is through emergence of the random matrix theory. However, as I will argue, this approach is rather related to the eigenstate thermalization hypothesis and ergodicity than to chaps. In this talk I will suggest that one can use fidelity susceptibility of equivalently geometric tensor and quantum Fisher information as a definition of chaos, which applies both to quantum and classical systems and which is related to long time tails of the auto-correlation functions of local perturbations. Through this approach we can establish of existence of the intermediate chaotic but non-ergodic regime separating integrable and ergodic phases, which have maximally sensitive eigenstates. I will discuss how this measure is also closely related to recently proposed definition of chaos through the Krylov complexity or the operator growth and that there is very interesting and still unexplained duality between short and long time behavior of chaotic and integrable systems As a specific example of this approach I will apply these ideas to interacting disordered systems and show that (many-body) localization is unstable in thermodynamic limit irrespective of the disorder strength.

## NSR singular vectors from Uglov polynomials

9 December in 11:30 (short)

Mikhail Bershtein

It was conjectured in 2012 that bosonization of a singular vector (in the Neveu–Schwarz sector) of the N=1 super analog of the Virasoro algebra can be identified with the Uglov symmetric function. We prove this conjecture. We also extend this result to the Ramond sector of the N=1 super-Virasoro algebra. Based on joint work with A, Vargulevich

## Long-range interactions between membrane inclusions: Electric field induced giant amplification of the pairwise potential

16 December in 11:30

E.S. Pikina, A.R. Muratov, E.I. Kats, V.V. Lebedev

The aim of this work is to revisit the phenomenological theory of the interaction between membrane inclusions, mediated by the membrane fluctuations. We consider the case where the inclusions are separated by distances larger than their characteristic size. Within our macroscopic approach a physical nature of such inclusions is not essential. However, we have always in mind two prototypes of such inclusions: proteins and RNA macromolecules. Because the interaction is driven by the membrane fluctuations and the coupling between inclusions and the membrane, it is possible to change the interaction potential by external actions affecting these factors. As an example of such external action we consider an electric field. Under external electric field (both dc or ac), we propose a new coupling mechanism between inclusions possessing dipole moments (as it is the case for most protein macromolecules) and the membrane. We found, quite unexpected and presumably for the first time, that the new coupling mechanism yields to giant enhancement of the pairwise potential of the inclusions. This result opens up a way to handle purposefully the interaction energy, and as well to test of the theory set forth in our article.
Results are published in Annals of Physics, in press, available online 11 May 2022, 168916; https://doi.org/10.1016/j.aop.2022.168916

## The structure of angular diagrams for systems describing the dynamics of an electron in a magnetic field for dispersion laws in general position

16 December in 11:30 (short)

I.A. Dynnikov, A.Ya. Maltsev, S.P. Novikov

We present a number of results that significantly refine the description of the angular diagrams that arise in the study of the dynamics of an electron in a magnetic field at all energy levels simultaneously. The description allows us to introduce some hierarchical structure on the set of stability zones on such diagrams, as well as to describe in more detail the set of occurrence of complex (chaotic) trajectories of the corresponding dynamical system. ZhETF, Volume 162, Issue. 2 (2022), UMN, volume 77, issue 6(468) (2022)

## Open level lines of a superposition of periodic potentials on a plane

16 December in 11:30 (short)

A.Ya. Maltsev, S.P. Novikov

We study the geometry of open potential level lines arising from the superposition of two different periodic potentials on a plane. This problem can be considered as a particular case of the Novikov problem on the behavior of open level lines of quasi-periodic potentials on a plane with four quasi-periods. At the same time, the formulation of this problem can have many additional features. We will give a general description of the emerging picture both in the most general case and in the presence of additional restrictions. The main approach to describing the behavior of open level lines is based on their division into topologically regular and chaotic level lines. Annals of Physics, In Press, Corrected Proof, Available online 22 July 2022, art. 169039; arXiv:2206.04014

## Interplay of superconductivity and localization near a 2D ferromagnetic quantum critical point

23 December in 11:30

P.A. Nosov, I.S. Burmistrov, S. Raghu

We study the superconducting instability of a two-dimensional disordered Fermi liquid weakly coupled to the soft fluctuations associated with proximity to an Ising-ferromagnetic quantum critical point. We derive interaction-induced corrections to the Usadel equation governing the superconducting gap function, and show that diffusion and localization effects drastically modify the interplay between fermionic incoherence and strong pairing interactions. In particular, we obtain the phase diagram, and demonstrate that: (i) there is an intermediate range of disorder strength where superconductivity is enhanced, eventually followed by a tendency towards the superconductor-insulator transition at stronger disorder; and (ii) diffusive particle-particle modes (so-called `Cooperons') acquire anomalous dynamical scaling z=4, indicating strong non-Fermi liquid behaviour.