We study reductions of the KdV equation and the Volterra lattice which correspond to stationary equations for the additional (non-commutative and non-local) symmetry subalgebra. In the general case, such reductions turn out to be equivalent to stationary equations for the sum of the Galilean or scaling symmetry and an arbitrary number of negative flows with different parameters. This brings them to a unified form of m-component systems of Painlevé type (continuous in the KdV case and discrete in the VL case). The corresponding isomonodromic Lax pairs and Bäcklund transformations forming the Z^m lattice are obtained.

The use of dissipation for the controlled creation of nontrivial quantum many-body correlated states is of much fundamental and practical interest. What is the result of imposing number conservation, which, in closed system, gives rise to diffusive spreading? We investigate this question for a paradigmatic model of a two-band system, with dissipative dynamics aiming to empty one band and to populate the other, which had been introduced before for the dissipative stabilization of topological states. Going beyond the mean-field treatment of the dissipative dynamics, we demonstrate the emergence of a diffusive regime for the particle and hole density modes at intermediate length- and time-scales, which, interestingly, can only be excited in nonlinear response to external fields. We also identify processes that limit the diffusive behavior of this mode at the longest length- and time-scales. Strikingly, we find that these processes lead to a reaction-diffusion dynamics governed by the Fisher-Kolmogorov-Petrovsky-Piskunov equation, making the designed dark state unstable towards a state with a finite particle and hole density.
Results are published in P. A. Nosov, D. S. Shapiro, M. Goldstein, I. S. Burmistrov, "Reaction-diffusive dynamics of number-conserving dissipative quantum state preparation", Phys. Rev. B 107, 174312 (2023)

Driven-dissipative protocols are proposed to control and create nontrivial quantum many-body correlated states. Protocols conserving the number of particles stand apart. As well-known, in quantum systems with the unitary dynamics the particle number conservation and random scattering yield diffusive behavior of two-particle excitations (diffusons and cooperons). Existence of diffusive modes in the particle-number-conserving dissipative dynamics is not well studied yet. We explicitly demonstrate the existence of diffusons in a paradigmatic model of a two-band system, with dissipative dynamics aiming to empty one fermion band and to populate the other one. The studied model is generalization of the model introduced in F. Tonielli, J. C. Budich, A. Altland, and S. Diehl, Phys. Rev. Lett. 124, 240404 (2020). We find how the diffusion coefficient depends on details of a model and the rate of dissipation. We discuss how the existence of diffusive modes complicates engineering of macroscopic many-body correlated states.
Results are published in A.A. Lyublinskaya, I.S. Burmistrov, "Diffusive modes of two-band fermions under number-conserving dissipative dynamics", Pis'ma v ZhETF 118, 538 (2023)

We study de Haas-van Alphen oscillations in a marginal Fermi liquid resulting from a three-dimensional metal tuned to a quantum-critical point (QCP). We show that the conventional approach based on extensions of the Lifshitz-Kosevich formula for the oscillation amplitudes becomes inapplicable when the correlation length exceeds the cyclotron radius. This breakdown is due to (i) non-analytic finite-temperature contributions to the fermion self-energy (ii) an enhancement of the oscillatory part of the self-energy by quantum fluctuations, and (iii) non-trivial dynamical scaling laws associated with the quantum critical point. We properly incorporate these effects within the Luttinger-Ward-Eliashberg framework for the thermodynamic potential by treating the fermionic and bosonic contributions on equal footing. As a result, we obtain the modified expressions for the oscillations of entropy and magnetization that remain valid in the non-Fermi liquid regime.

Generalized multifractality characterizes system size dependence of pure scaling local observables at Anderson transitions in all ten symmetry classes of disordered systems. Here we demonstrate that the concept of generalized multifractality can be extended to local observables situated neat boundaries of critical disordered noninteracting systems. We study the generalized boundary multifractality focusing on the spin quantum Hall symmetry class (class C). Employing the two-loop renormalization group analysis within Finkel'stein nonlinear sigma model we compute analytically the anomalous dimensions of the pure scaling operators located at the boundary of the system. We find that generalized boundary multifractal exponents are twice larger than their bulk counterparts. Also, in two dimensions we compute the corresponding boundary multifractal exponents numerically.
A talk is based on the following papers
1. S.S. Babkin, J.F. Karcher, I.S. Burmistrov, A.D. Mirlin, "Generalized surface multifractality in 2D disordered systems", Phys. Rev. B 108, 104205 (2023)
2. S.S. Babkin, I.S. Burmistrov, "Boundary multifractality in the spin quantum Hall symmetry class with interaction", arxiv:2308.16852

We investigate theoretically and numerically the dynamics of long-living bound state coherent structures, namely bi-solitons, obtained earlier in [1] in the framework of the Zakharov equation and the exact nonlinear RV-equations. To elucidate the long-living bi-soliton complex nature we propose a semi-analytical approach based on the perturbation theory and inverse scattering transform (IST) for the 1D focusing nonlinear Schrödinger equation (NLSE). We present the Zakharov equation and the RV-equations as the NLSE plus a right-hand side in order to apply our approach. Then we compute the IST scattering data for a time series of the bi-soliton wavefield, and observe a periodic energy exchange between two solitons and continuous spectrum radiation resulting in stable oscillations of the coherent structure. We find that soliton eigenvalues oscillate on stable trajectories experiencing a slight drift on a scale of hundreds of oscillation periods. In addition, after obtaining the change of the bi-soliton eigenvalues, we observe that they are in good agreement with predictions of the IST perturbation theory. Based on these results we conclude that the IST perturbation theory justifies the existence of the bound state coherent structures on the surface of deep water which emerge as a result of a balance between the dominant solitonic part and a portion of continuous spectrum radiation.
[1]. Kachulin, D., Dremov, S., Dyachenko, A. (2021). Bound coherent structures propagating on the free surface of deep water. Fluids, 6(3), 115.

We suggest a new computer-assisted approach to the development of turbulence theory. It allows one to impose lower and upper bounds on correlation functions using sum-of-squares polynomials. We demonstrate it on the minimal cascade model of two resonantly interacting modes, when one is pumped and the other dissipates. We show how to present correlation functions of interest as part of a sum-of-squares polynomial using the stationarity of the statistics. That allows us to find how the moments of the mode amplitudes depend on the degree of non-equilibrium (analog of the Reynolds number), which reveals some properties of marginal statistical distributions. By combining scaling dependence with the results of direct numerical simulations, we obtain the probability densities of both modes in a highly intermittent inverse cascade. We also show that the relative phase between modes tends to π/2 and -π/2 in the direct and inverse cascades as the Reynolds number tends to infinity, and derive bounds on the phase variance. Our approach combines computer-aided analytical proofs with a numerical algorithm applied to high-degree polynomials.
Phys. Rev. E 107, 054114 (2023); arXiv:2302.03757

Chromosomes are exceedingly long topologically-constrained polymers compacted in a cell nucleus. We recently suggested that chromosomes are organized into loops by an active process of loop extrusion. Yet loops remain elusive to direct observations in living cells; detection and characterization of myriads of such loops is a major challenge. The lack of a tractable physical model of a polymer folded into loops limits our ability to interpret experimental data and detect loops. Here, we introduce a new physical model – a polymer folded into a sequence of loops, and solve it analytically. Our model shows how loops affect statistics of contacts in a polymer across different scales, explaining universally observed shapes of the contact probability. Moreover we analyze how folding into loops affects topological properties of crumpled polymers.

I will discuss some aspects of the Born-Oppenheimer potentials for doubly heavy pentaquarks (quarks systems with two heavy and three light quarks/antiquarks).