arXiv subjects

Mykola Semenyakin

Publications and source records attributed to Mykola Semenyakin.

7 recordsLinked to original sources

arXiv2025

Classifying fermionic states via many-body correlation measures

Understanding the structure of quantum correlations in a many-body system is key to its computational treatment. For fermionic systems, correlations can be defined as deviations from Slater determinant states. The link between fermionic correlations and efficient computational physics methods is actively studied but remains ambiguous. We make progress in establishing this connection mathematically. In particular, we find a rigorous classification of states relative to $k$-fermion correlations, which admits a computational physics interpretation. Correlations are captured by a measure $ω_k$, a function of $k$-fermion reduced density matrix that we call twisted purity. A condition $ω_k=0$ for a given $k$ puts the state in a class $G_k$ of correlated states. Sets $G_k$ are nested in $k$, and Slater determinants correspond to $k = 1$. Classes $G_{k=O(1)}$ are shown to be physically relevant, as $ω_k$ vanishes or nearly vanishes for truncated configuration-interaction states, perturbation series around Slater determinants, and some nonperturbative eigenstates of the 1D Hubbard model. For each $k = O(1)$, we give an explicit ansatz with a polynomial number of parameters that covers all states in $G_k$. Potential applications of this ansatz and its connections to the coupled-cluster wavefunction are discussed.

quant-ph↗

arXiv2024

A brief introduction to quantum groups

These are lecture notes of a mini-course given by the first author in Moscow in July 2019, taken by the second author and then edited and expanded by the first author. They were also a basis of the lectures given by the first author at the CMSA Math Science Literature Lecture Series in May 2020. We attempt to give a bird's-eye view of basic aspects of the theory of quantum groups.

math.QA↗

arXiv2024

Cluster Reductions, Mutations, and $q$-Painlevé Equations

We propose an extension of the Goncharov-Kenyon class of cluster integrable systems by their Hamiltonian reductions. This extension allows us to fill in the gap in cluster construction of the $q$-difference Painlevé equations, showing that all of them can be obtained as deautonomizations of the reduced Goncharov-Kenyon systems. Conjecturally, the isomorphisms of reduced Goncharov-Kenyon integrable systems are given by mutations in another, dual in some sense, cluster structure. These are the polynomial mutations of the spectral curve equations and polygon mutations of the corresponding decorated Newton polygons. In the Painlevé case the initial and dual cluster structures are isomorphic. It leads to self-duality between the spectral curve equation and the Painlevé Hamiltonian, and also extends the symmetry from affine to elliptic Weyl group.

nlin.SI↗

arXiv2022

Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux

Important illustration to the principle ``partition functions in string theory are $τ$-functions of integrable equations'' is the fact that the (dual) partition functions of $4d$ $\mathcal{N}=2$ gauge theories solve Painlevé equations. In this paper we show a road to self-consistent proof of the recently suggested generalization of this correspondence: partition functions of topological string on local Calabi-Yau manifolds solve $q$-difference equations of non-autonomous dynamics of the ``cluster-algebraic'' integrable systems. We explain in details the ``solutions'' side of the proposal. In the simplest non-trivial example we show how $3d$ box-counting of topological string partition function appears from the counting of dimers on bipartite graph with the discrete gauge field of ``flux'' $q$. This is a new form of topological string/spectral theory type correspondence, since the partition function of dimers can be computed as determinant of the linear $q$-difference Kasteleyn operator. Using WKB method in the ``melting'' $q\to 1$ limit we get a closed integral formula for Seiberg-Witten prepotential of the corresponding $5d$ gauge theory. The ``equations'' side of the correspondence remains the intriguing topic for the further studies.

hep-th↗

arXiv2021

Solution of tetrahedron equation and cluster algebras

We notice a remarkable connection between Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism we show how to construct integrable system with spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to general non-symmetric Newton polygons, and prove Lemma, which classifies conjugacy classes in double affine Weyl groups of $A$-type by Newton polygons.

nlin.SI↗

arXiv2017

Alternating currents and shear waves in viscous electronics

Strong interaction among charge carriers can make them move like viscous fluid. Here we explore alternating current (AC) effects in viscous electronics. In the Ohmic case, incompressible current distribution in a sample adjusts fast to a time-dependent voltage on the electrodes, while in the viscous case, momentum diffusion makes for retardation and for the possibility of propagating slow shear waves. We focus on specific geometries that showcase interesting aspects of such waves: current parallel to a one-dimensional defect and current applied across a long strip. We find that the phase velocity of the wave propagating along the strip respectively increases/decreases with the frequency for no-slip/no-stress boundary conditions. This is so because when the frequency or strip width goes to zero (alternatively, viscosity go to infinity), the wavelength of the current pattern tends to infinity in the no-stress case and to a finite value in a general case. We also show that for DC current across a strip with no-stress boundary, there only one pair of vortices, while there is an infinite vortex chain for all other types of boundary conditions.

cond-mat.mes-hall↗

arXiv2014

On diagrammatic technique for nonlinear dynamical systems

In this paper we investigate phase flows over $\mathbb{C}^n$ and $\mathbb{R}^n$ generated by vector fields $V=\sum P^{i}\partial_i$ where $P^{i}$ are finite degree polynomials. With the convenient diagrammatic technique we get expressions for evolution operators $ev\{V|t\}: x(0)\mapsto x(t)$ through the series in powers of $x(0)$ and $t$, represented as sum over all trees of particular type. Estimates are made for the radius of convergence in some particular cases. The phase flows behavior in the neighborhood of vector field fixed points are examined. Resonance cases are considered separately.

math-ph↗