scholarly journals Quantum K-theory of quiver varieties and many-body systems

2021 ◽  
Vol 27 (5) ◽  
Author(s):  
Peter Koroteev ◽  
Petr P. Pushkar ◽  
Andrey V. Smirnov ◽  
Anton M. Zeitlin
Keyword(s):  
Toda Lattice ◽  
Type A ◽  
Spin Chains ◽  
Flag Varieties ◽  
Quantum Geometry ◽  
Many Body ◽  
Quiver Varieties ◽  
K Theory ◽  
Many Body Systems ◽  
Theoretic Version

AbstractWe define quantum equivariant K-theory of Nakajima quiver varieties. We discuss type A in detail as well as its connections with quantum XXZ spin chains and trigonometric Ruijsenaars-Schneider models. Finally we study a limit which produces a K-theoretic version of results of Givental and Kim, connecting quantum geometry of flag varieties and Toda lattice.

scholarly journals BYPASSING STATE INITIALIZATION IN HAMILTONIAN TOMOGRAPHY ON SPIN-CHAINS

2011 ◽  
Vol 09 (supp01) ◽  
pp. 181-187 ◽  
Author(s):  
C. DI FRANCO ◽  
M. PATERNOSTRO ◽  
M. S. KIM
Keyword(s):  
Spin Chain ◽  
Spin Chains ◽  
Chain Model ◽  
Many Body ◽  
Extensive Discussion ◽  
Physics Community ◽  
Spin Chain Model ◽  
Many Body Systems ◽  
Quantum Protocol

We provide an extensive discussion on a scheme for Hamiltonian tomography of a spin-chain model that does not require state initialization [Phys. Rev. Lett.102 (2009) 187203]. The method has spurred the attention of the physics community interested in indirect acquisition of information on the dynamics of quantum many-body systems and represents a genuine instance of a control-limited quantum protocol.

Introduction to Integrable Many-Body Systems II

2010 ◽  
Vol 60 (2) ◽  
Author(s):  
Ladislav Šamaj
Keyword(s):  
Bethe Ansatz ◽  
Ground State ◽  
Quantum Spin ◽  
Spin Chains ◽  
Scattering Method ◽  
Heisenberg Chain ◽  
Many Body ◽  
Xxz Model ◽  
The Subject ◽  
Many Body Systems

Introduction to Integrable Many-Body Systems IIThis is the second part of a three-volume introductory course about integrable systems of interacting bodies. The models of interest are quantum spin chains with nearest-neighbor interactions between spin operators, in particular Heisenberg spin-1/2 models. The Ising model in a transverse field, expressible as a quadratic fermion form by using the Jordan-Wigner transformation, is the subject of Sect. 12. The derivation of the coordinate Bethe ansatz for the XXZ Heisenberg chain and the determination of its absolute ground state in various regions of the anisotropy parameter are presented in Sect. 13. The magnetic properties of the ground state are explained in Sect. 14. Sect. 15 concerns excited states and the zero-temperature thermodynamics of the XXZ model. The thermodynamics of the XXZ Heisenberg chain is derived on the basis of the string hypothesis in Sect. 16; the thermodynamic Bethe ansatz equations are analyzed in high-temperature and low-temperature limits. An alternative derivation of the thermodynamics without using strings, leading to a non-linear integral equation determining the free energy, is the subject of Sect. 17. A nontrivial application of the Quantum Inverse Scattering method to the fully anisotropic XYZ Heisenberg chain is described in Sect. 18. Sect. 19 deals with integrable cases of isotropic spin chains with an arbitrary spin.

scholarly journals Decompositions of Grothendieck Polynomials

2017 ◽  
Vol 2019 (10) ◽  
pp. 3214-3241 ◽  
Author(s):  
Oliver Pechenik ◽  
Dominic Searles
Keyword(s):  
Type A ◽  
General Context ◽  
Flag Varieties ◽  
Combinatorial Formula ◽  
Structure Constants ◽  
Schubert Classes ◽  
K Theory ◽  
Standing Problem ◽  
Grothendieck Polynomials

AbstractWe investigate the long-standing problem of finding a combinatorial rule for the Schubert structure constants in the $K$-theory of flag varieties (in type $A$). The Grothendieck polynomials of A. Lascoux–M.-P. Schützenberger (1982) serve as polynomial representatives for $K$-theoretic Schubert classes; however no positive rule for their multiplication is known in general. We contribute a new basis for polynomials (in $n$ variables) which we call glide polynomials, and give a positive combinatorial formula for the expansion of a Grothendieck polynomial in this basis. We then provide a positive combinatorial Littlewood–Richardson rule for expanding a product of Grothendieck polynomials in the glide basis. Our techniques easily extend to the $\beta$-Grothendieck polynomials of S. Fomin–A. Kirillov (1994), representing classes in connective $K$-theory, and we state our results in this more general context.

scholarly journals Breaking of Huygens-Fresnel principle in inhomogeneous Tomonaga-Luttinger liquids

2021 ◽  
Author(s):  
Marek Gluza ◽  
Per Moosavi ◽  
Spyros Sotiriadis
Keyword(s):  
Light Cone ◽  
Ultracold Atoms ◽  
Spin Chains ◽  
Many Body ◽  
Luttinger Liquids ◽  
Main Motivation ◽  
One Dimensional ◽  
Superconducting Circuits ◽  
Many Body Systems ◽  
Two Parameters

Abstract Tomonaga-Luttinger liquids (TLLs) can be used to effectively describe one-dimensional quantum many-body systems such as ultracold atoms, charges in nanowires, superconducting circuits, and gapless spin chains. Their properties are given by two parameters, the propagation velocity and the Luttinger parameter. Here we study inhomogeneous TLLs where these are promoted to functions of position and demonstrate that they profoundly affect the dynamics: In general, besides curving the light cone, we show that propagation is no longer ballistically localized to the light-cone trajectories, different from standard homogeneous TLLs. Specifically, if the Luttinger parameter depends on position, the dynamics features pronounced spreading into the light cone, which cannot be understood via a simple superposition of waves as in the Huygens-Fresnel principle. This is the case for ultracold atoms in a parabolic trap, which serves as our main motivation, and we discuss possible experimental observations in such systems.

REGULAR STRUCTURE OF LOW-LYING STATES FOR ATOMIC NUCLEI UNDER RANDOM INTERACTIONS

2008 ◽  
Vol 17 (supp01) ◽  
pp. 304-317
Author(s):  
Y. M. ZHAO
Keyword(s):  
Ground State ◽  
Collective Motion ◽  
Regular Structure ◽  
Positive Parity ◽  
Many Body ◽  
Random Interactions ◽  
Atomic Nuclei ◽  
Many Body Systems

In this paper we review regularities of low-lying states for many-body systems, in particular, atomic nuclei, under random interactions. We shall discuss the famous problem of spin zero ground state dominance, positive parity dominance, collective motion, odd-even staggering, average energies, etc., in the presence of random interactions.

scholarly journals Emergence of a Renormalized 1/N Expansion in Quenched Critical Many-Body Systems

2021 ◽  
Vol 126 (11) ◽  
Author(s):  
Benjamin Geiger ◽  
Juan Diego Urbina ◽  
Klaus Richter
Keyword(s):  
Many Body ◽  
Many Body Systems

scholarly journals Continuous Phase Transition without Gap Closing in Non-Hermitian Quantum Many-Body Systems

2020 ◽  
Vol 125 (26) ◽  
Author(s):  
Norifumi Matsumoto ◽  
Kohei Kawabata ◽  
Yuto Ashida ◽  
Shunsuke Furukawa ◽  
Masahito Ueda
Keyword(s):  
Phase Transition ◽  
Continuous Phase ◽  
Many Body ◽  
Continuous Phase Transition ◽  
Many Body Systems ◽  
Gap Closing
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