arXiv subjects

Sikarin Yoo-Kong

Publications and source records attributed to Sikarin Yoo-Kong.

31 records · Page 2Linked to original sources

arXiv2022

The multi-time propagators and the consistency condition

The time evolution of a wave function with $N$ time variables through the Feynman picture of quantum mechanics is derived. However, these evolutions will be compatible if and only if the $N$ Lagrangians satisfy a certain relation called the consistency condition or integrability condition which could be expressed in terms of the Wilson line. This consistency condition violates if there presents the interaction. As a consequence of this consistency condition, the evolution of the wave function gives rise to a key feature called the "path-independent" property on the space of time variables. This would suggest that one must consider all possible paths not only on the space of dependent variables but also on the space of independent variables. In the view of the geometry, this consistency condition can be considered as a zero curvature condition and the multi-time evolution can be treated as a compatible parallel transport on flat space of time variables.

quant-ph↗

arXiv2020

Geodesic compatibility: Goldfish systems

To capture a multidimensional consistency feature of integrable systems in terms of the geometry, we give a condition called \emph{geodesic compatibility} that implies the existence of integrals in involution of the geodesic flow. The geodesic compatibility condition is constructed from a concrete example namely the integrable Calogero's Goldfish system through the Poisson structure and the variational principle. The geometrical view of the geodesic compatibility gives a compatible parallel transport between two different Hamiltonian vector fields.

nlin.SI↗

arXiv2020

Hamiltonian Zoo for systems with one degree of freedom

We present alternative explicit forms of the standard Hamiltonian for systems with one degree of freedom. This new class of infinite Hamiltonians is called Newton-equivalent Hamiltonian zoo, producing the same equation of motion. These Hamiltonians are directly solved from the Hamilton's equations and come with extra-parameters, which are interpreted as scaling factors for the time evolution on phase space. Moreover, each Hamiltonian in the zoo can be used as a generating function for a Hamiltonian hierarchy.

math-ph↗

arXiv2019

Integrable Hamiltonian Hierarchies and Lagrangian 1-Forms

We present further developments on the Lagrangian 1-form description for one-dimensional integrable systems in both discrete and continuous levels. A key feature of integrability in this context called a closure relation will be derived from the local variation of the action on the space of independent variables. The generalised Euler-Lagrange equations and constraint equations are derived directly from the variation of the action on the space of dependent variables. This set of Lagrangian equations gives rise to a crucial property of integrable systems known as the multidimensional consistency. Alternatively, the closure relation can be obtained from generalised Stokes' theorem exhibiting a path independent property of the systems on the space of independent variables. The homotopy structure of paths suggests that the space of independent variables is simply connected. Furthermore, the Nöether charges, invariants in the context of Liouville integrability, can be obtained directly from the non-local variation of the action on the space of dependent variables.

math-ph↗

arXiv2017

Ground state entanglement entropy for discrete-time two coupled harmonic oscillators

The ground state entanglement of the system, both in discrete-time and continuous-time cases, is quantified through the linear entropy. The result shows that the entanglement increases as the interaction between the particles increases in both time scales. It is also found that the strength of the harmonic potential affects the formation rate of the entanglement of the system. The different feature of the entanglement between continuous-time and discrete-time scales is that, for discrete-time entanglement, there is a cut-off condition. This condition implies that the system can never be in a maximally entangled state.

math-ph↗

arXiv2017

The multiplicative Hamiltonian and its hierarchy

The multiplicative Hamiltonian flow on the phase space for a system with 1 degree of freedom was constituted from infinite hierarchy Hamiltonian flows. A new type of canonical transformation associated with the multiplicative Hamiltonian was found and called the λ- extended class of the standard canonical transformations.

math-ph↗

arXiv2016

On the multiplicative form of the Lagrangian

An alternative class of the Lagrangian called the multiplicative form is suc- cessfully derived for a system with one degree of freedom for both non-relativistic and relativistic cases. This new Lagrangian can be considered as a 1-parameter: λ extended class from the standard additive form of the Lagrangian since both yield the same equation of motion. Remarkably, the multiplicative form of the Lagrangian could be treated as a generating function to produce an infinite hierar- chy of Lagrangians which give the same equation of motion. This nontrivial set of Lagrangians confirms that indeed Lagrange function is not unique.

math-ph↗

arXiv2016

On the Lagrangian 1-Form Structure of the Hyperbolic Calogero-Moser System

In this work, we present another example of the Lagrangian 1-form structure for the hy- perbolic Calogero-Moser system both in discrete-time level and continuous-time level. The discrete-time hyperbolic Calogero-Moser system is obtained by considering pole-reduction of the semi-discrete Kadomtsev-Petviashvili (KP) equation. The key relation called the discrete-time closure relation is directly obtained from the compatibility between the temporal Lax matrices. The continuous-time hierarchy of the hyperbolic Calogero-Moser system is obtained through two successive continuum limits. The continuous-time closure relation, which is a consequence of continuum limits on the discrete-time one, is also shown to hold.

nlin.SI↗

arXiv2015

Double path-integral method for obtaining the mobility of the one-dimensional charge transport in molecular chain

We report on a theoretical investigation concerning the polaronic effect on the transport properties of a charge carrier in the one-dimensional molecular chain. Our technique is based on the Feynman's path integral approach. Analytical expressions for the frequency-dependent mobility and effective mass of the carrier are obtained as functions of electron-phonon coupling. The result exhibits the crossover from a nearly free particle to a heavily trapped particle. We find that the mobility depends on temperature and decreases exponentially with increasing temperature at low temperature. It exhibits large-polaronic-like behaviour in the case of weak electron-phonon coupling. These results agree with the phase transition \cite{Mish} of transport phenomena related to polaron motion in the molecular chain.

cond-mat.mes-hall↗

arXiv2014

On the Lagrangian structure of Calogero's Goldfish model

The discrete-time rational Calogero's goldfish system is obtained from the Ansatz Lax pair. The discrete-time Lagrangians of the system possess the discrete-time 1-form structure as those in the discrete-time Calogero-Moser system and discrete-time Ruijsenaars-Schneider system. Performing two steps of continuum limits, we obtain Lagrangian hierarchy for the system. Expectingly, the continuous-time Lagrange 1-form structure of the system holds. Furthermore, the connection to the lattice KP systems is also established.

nlin.SI↗

arXiv2014

Entanglement entropy for a particle coupled with its surrounding

We investigate the entanglement for a model of a particle moving in the lattice (many-body system). The interaction between the particle and the lattice is modelled using Hooke's law. The Feynman path integral approach is applied to compute the density matrix of the system. The complexity of the problem is reduced by considering two-body system (bipartite system). The spatial entanglement of ground state is studied using the linear entropy. We find that increasing the confining potential implies a large spatial separation between the two particles. Thus the interaction between the particles increases according to Hooke's law. This results in the increase in the spatial entanglement.

math-ph↗

arXiv2013

Discrete-time Ruijsenaars-Schneider system and Lagrangian 1-form structure

We study the Lagrange formalism of the (rational) Ruijsenaars-Schneider (RS) system, both in discrete time as well as in continuous time, as a further example of a Lagrange 1-form structure in the sense of the recent paper [24]. The discrete-time model of the RS system was established some time ago arising via an Ansatz of a Lax pair, and was shown to lead to an exactly integrable correspondence (multivalued map)[15]. In this paper we consider an extended system representing a family of commuting flows of this type, and establish a connection with the lattice KP system. In the Lagrangian 1-form structure of this extended model, the closure relation is verified making use of the equations of motion. Performing successive continuum limits on the RS system, we establish the Lagrange 1-form structure for the corresponding continuum case of the RS model.

nlin.SI↗

arXiv2011

Elliptic (N,N^\prime)-Soliton Solutions of the lattice KP Equation

Elliptic soliton solutions, i.e., a hierarchy of functions based on an elliptic seed solution, are constructed using an elliptic Cauchy kernel, for integrable lattice equations of Kadomtsev-Petviashvili (KP) type. This comprises the lattice KP, modified KP (mKP) and Schwarzian KP (SKP) equations as well as Hirota's bilinear KP equation, and their successive continuum limits. The reduction to the elliptic soliton solutions of KdV type lattice equations is also discussed.

nlin.SI↗