Commensurate-incommensurate phase transition in one-dimensional quantum sine-Gordon model based on a lattice massive thirring model

The commensurate-incommensurate (C-IC) phase transition in the one dimensional quantum sine-Gordon model at zero temperature is exactly solved with the use of the Bethe ansatz technique for the lattice massive Thirring model. The energy difference between C and IC phases is derived based on the same ground state which is valid in the whole parameter region. It is due to the fact that there is no change in the ground state of the lattice massive Thirring model even in the strongly repulsive region in contrast to the continuum massive Thirring model even in the strongly repulsive region in contrast to the continuum massive Thirring model. It is proved in the whole parameter region that the IC phase can be realized with the soliton density proportional to (E _s : formation energy of soliton), whenE _s becomes negative.     

The cluster expansion for classical and quantum lattice systems

We develop a high-temperature expansion for general lattice systems which can be applied to classical as well as quantum systems. Applying the expansion we prove analyticity of correlation functions, uniqueness of equilibrium states, and cluster properties for classical and quantum lattice systems in the high-temperature region.     

Quantum discrete sine-Gordon model at roots of 1: Integrable quantum system on the integrable classical background

The quantum discrete sine-Gordon model at roots of 1 is studied. It is shown that this model provides an example of an integrable quantum system in an integrable classical background. In particular, the spectrum of quantum integrals of motions in this model depends only on the values of integrals of motion of a background classical system.Dedicated to L.D. Faddeev on his 60th birthday     

The Weyl representation in classical and quantum mechanics

The position representation of the evolution operator in quantum mechanics is analogous to the generating function formalism of classical mechanics. Similarly, the Weyl representation is connected to new generating functions described by chords and centres in phase space. Both classical and quantal theories relie on the group of translations and reflections through a point in phase space. The composition of small time evolutions leads to new versions of the classical variational principle and to path integrals in quantum mechanics. The strong resemblance between the two theories allows a clear derivation of the semiclassical limit in which observables evolve classically in the Weyl representation. The restriction of the motion to the energy shell in classical mechanics is the basis for a full review of the semiclassical Wigner function and the theory of scars of periodic orbits. By embedding the theory of scars in a fully uniform approximation, it is shown that the region in which the scar contribution is oscillatory is separated from a decaying region by a caustic that touches the shell along the periodic orbit and widens quadratically within the energy shell.     

Diagonal representation for a generic matrix valued quantum Hamiltonian

A general method to derive the diagonal representation for a generic matrix valued quantum Hamiltonian is proposed. In this approach new mathematical objects like non-commuting operators evolving with the Planck constant promoted as a running variable are introduced. This method leads to a formal compact expression for the diagonal Hamiltonian which can be expanded in a power series of the Planck constant. In particular, we provide an explicit expression for the diagonal representation of a generic Hamiltonian to the second order in the Planck constant. This result is applied, as a physical illustration, to Dirac electrons and neutrinos in external fields.     

Separating the classical and quantum information via quantum cloning

An application of quantum cloning to optimally interface a quantum system with a classical observer is presented; in particular, we describe a procedure to perform a minimal disturbance measurement on a single qubit by adopting a 1-->2 cloning machine followed by a generalized measurement on a single clone and the anticlone or on the two clones. Such a scheme can be applied to enhance the transmission fidelity over a lossy quantum channel.     

The lattice quantum Sine-Gordon model

The integrable statistical physics model on the rectangular two-dimensional lattice which we call the L-model is constructed. This model generates the integrable quantum sine-Gordon model on the one-dimensional lattice in the same way as the ice model generates the XXZ model.     

Thermodynamics of the quantum sine-Gordon model with finite winding number

Thermodynamics of the quantum sine-Gordon model is formulated in the presence of finite winding number. Several asymptotic cases are studied analytically.     

Relativistic quantum and classical kinetic equations in the tomographic-probability representation

Using the Radon integral transform of the relativistic kinetic equation for a spin-zero particle, we obtain the classical and quantum evolution equations for the tomographic probability density (tomogram) describing the states of the particle in both the classical and quantum pictures. The Green functions (propagators) of the evolution equations of a free particle are constructed. The examples of the evolution of Gaussian tomogram is considered.     

Estimates of critical lengths and critical temperatures for classical and quantum lattice systems

Local Ward identities are derived which lead to the mean-field upper bound for the critical temperature for certain multicomponent classical lattice systems (improving by a factor of two an estimate of Brascamp-Lieb). We develop a method for accurately estimating lattice Green's functionsI _d yielding 0.3069<I ₄<0.3111 and the global bounds (d?1/2)^?<I _d <(d?1)^? for alld?4. The estimate forI _d implies the existence of a critical length for classical lattice systems with fixed length spins. Forv-component spins with fixed lengthb on the lattice ? ^d ,v=1, 2, 3, 4, the critical temperature for spontaneous magnetization satisfies $$\frac{{2Jb^2 }}{k}\frac{{d - 1}}{v}< T{}_c \leqslant \frac{{2Jb^2 }}{k}\frac{d}{v} for d \geqslant 4$$ ford4 Using GHS or generalized Griffiths' inequalities, we find that the upper bounds on the critical temperature extend to certain classical and quantum systems with unbounded spins. Absence of symmetry breakdown at high temperature for quantum lattice fields follows from bounding the energy density by a multiple ofkT. Path space techniques for finite degrees of freedom show that the high-temperature limit is classical.     

Critical values of the Yang-Yang functional in the quantum sine-Gordon model

The critical values of the Yang-Yang functional corresponding to the vacuum states of the sine-Gordon QFT in the finite-volume are studied. Two major applications are discussed: (i) generalization of Fendley-Saleur-Zamolodchikov relations to arbitrary values of the sine-Gordon coupling constant, and (ii) connection problem for a certain two-parameter family of solutions of the Painlevé III equation.     

Time-dependent variational method for sine-Gordon quantum field theory

The sine-Gordon model in 1+1 dimensions is studied within the Schrödinger framework for field theory. In particular we evaluate the effective potential and examine the finiteness ofm(t), the soliton mass, for allt.     

Bethe lattice representation for sandpiles

Avalanches in sandpiles are represented by a process of percolation in a Bethe lattice with a feedback mechanism. The results indicate that the frequency spectrum and probability distribution of avalanches provide a better resemblance to the experimental results than other models using cellular automata simulations. Apparent discrepancies between experiments performed by different authors are reconciled. Critical behavior is expressed here by the critical properties of percolation phenomena.     

Solvable lattice models related to the vector representation of classical simple Lie algebras

A series of solvable lattice models with face interaction are introduced on the basis of the affine Lie algebraX _n ⁽¹⁾ =A _n ⁽¹⁾ ,B _n ⁽¹⁾ ,C _n ⁽¹⁾ ,D _n ⁽¹⁾ . The local states taken on by the fluctuation variables are the dominant integral weights ofX _n ⁽¹⁾ of a fixed level. Adjacent local states are subject to a condition related to the vector representation ofX _n . The Boltzmann weights are parametrized by elliptic theta functions and solve the star-triangle relation.     

A probabilistic model of quantum states for classical data security

The phenomenal progress of quantum information theory over the last decade has substantially broadened the potential to simulate the superposition of states for exponential speedup of quantum algorithms over their classical peers. Therefore, the conventional and modern cryptographic standards (encryption and authentication) are susceptible to Shor’s and Grover’s algorithms on quantum computers. The significant improvement in technology permits consummate levels of data protection by encoding classical data into small quantum states that can only be utilized once by leveraging the capabilities of quantum-assisted classical computations. Considering the frequent data breaches and increasingly stringent privacy legislation, we introduce a hybrid quantum-classical model to transform classical data into unclonable states, and we experimentally demonstrate perfect state transfer to exemplify the classical data. To alleviate implementation complexity, we propose an arbitrary quantum signature scheme that does not require the establishment of entangled states to authenticate users in order to transmit and receive arbitrated states to retrieve classical data. The consequences of the probabilistic model indicate that the quantum-assisted classical framework substantially enhances the performance and security of digital data, and paves the way toward real-world applications.     

A one-dimensional lattice model for a quantum mechanical free particle

Two types of particles, A and B with their corresponding antiparticles, are defined in a onedimensional cyclic lattice with an odd number of sites. In each step of time evolution, each particle acts as a source for the polarization field of the other type of particle with nonlocal action but with an effect decreasing with the distance: . It is shown that the combined distribution of these particles obeys the time evolution of a free particle as given by quantum mechanics.     

Sharing of classical and quantum correlations via XY interaction

The sharing of classical and quantum correlations via XY interaction is investigated. The model includes two identical networks consisting of n$n$ nodes, the i$i$th node of one network sharing a correlated state with the j$j$th node of the other network, while all other nodes are initially unconnected. It is shown that classical correlation, quantum discord as well as entanglement can be shared between any two nodes of the network via XY interaction and that quantum information can be transferred effectively between them. It is found that there is no simple dominating relation between the quantum correlation and entanglement in inertial system.