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April 1, 2019 | 2:45 p.m. - 4:00 p.m.
Category: Lecture
Location: Faculty/Administration #1140 | Map
656 W. Kirby
Detroit, MI 48202
Cost: Free

Speaker: Vladimir Chernyak, Wayne State University (Chemistry)

Title:

Integrability in Time-Dependent Quantum Problems: Factorization, Moduli Spaces, Spectral Curves, and Representations of Quantum Groups

Abstract:

Quantum evolution with time-dependent Hamiltonians, $ihbar dot{Psi}(t) = {H}(t) Psi(t)$, as of today draws considerable attention, both in experimental and theoretical research. The simplest model with ${H}(t) = {A} + {B}t$,  ${A}$ and ${B}$ being $2times 2$ real hermitian matrices, known as the Landau-Zener (LZ) problem, has an exact solution in special functions. In the general $N$-dimensional case, known as Multi-Level LZ (MLZ) problem, exact solutions are not available. However, for a certain class of MLZ problems that satisfy certain phenomenologically determined ``integrability'' conditions, the scattering matrix can be represented in a factorized form, with the elementary scattering events being represented in terms of the standard LZ matrix, which is the first aspect of integrability.
 
In the first part of this talk we reveal the reason that stands behind the aforementioned factorization: Each integrable MLZ problem can be embedded into a system of $M$ linear first-order differential equations with respect to $M$-dimensional time, that satisfy the consistency constraints, the latter having a form of the zero-curvature conditions, which is the second and dynamical aspect of integrability. In other words, the meromorphic connection/$D$-module over $CP^{1}$ is a pull-back of some counterpart, defined on $CP^{M}$.
 
In the second part of the talk we will discuss the problem of identification of exactly solvable MLZ problems and show that it boils down to classification of meromorphic connections (as special kinds of $D$-modules) over $CP^{M}$, equipped with additional structures, associated with time-reversal symmetry, possessed by MLZ problems. The corresponding moduli spaces turn out to be quotient moduli stacks. Since with each MLZ problem one can associate a smooth complex curve, there is a morphism from the classifying stack of integrable MLZ to the moduli stack of complex curves. We will speculate on possible relevance to number theory, and in particular modular forms.
 
In the last part of the talk we focus on a more general, still integrable, so-called BCS model that describes $N_{rm s}$ identical quantum spins, with $H(t) = H_{rm C} t^{-1} + H_{rm G}$, where $H_{rm C}$ and $H_{rm G}$ describe identical pairwise spin-spin interactions and effects of inhomogeneous magnetic field, respectively, so that the model describes evolution of the initial strongly correlated state at $t = +0$ into an uncorrelated counterpart at $t to infty$, when the spin-spin interaction is switched of, and generally in a non-adiabatic fashion. The obtained exact solution shows an amazing property: The ground state in the strongly correlated phase ``dissociates'' into a Gibbs distribution in a completely integrable way, without any chaos or bath involved. Evolution of an arbitrary initial eigenstate still requires solving a complex combinatorial problem. It can be efficiently treated by making use of the spin permutation symmetry, with the symmetry of the extended model that provides exact solutions being extended from the symmetric group ${rm S}_{N_{rm s}}$ to Artin's Braid Group (BG) ${rm B}_{N_{rm s}}$, due to effects of monodromy in the extended space. The compact Quantum Group (QG) ${rm SU}_{q} (2)$ appears naturally as an auxiliary tool for describing relevant representation of ${rm B}_{N_{rm s}}$.

 

For more information about this event, please contact Department of Mathematics at 3135772479 or math@wayne.edu.