Leading large N modification of QCD2 on a cylinder by dynamical fermions

We consider two-dimensional QCD on a cylinder, where space is a circle. We find the ground state of the system in case of massless quarks in a 1/Nexpansion. We find that coupling to fermions nontrivially modifies the large N saddle point of the gauge theory due to the phenomenon of “decompactification” of eigenvalues of the gauge field. We calculate the vacuum energy and the vacuum expectation value of the Wilson loop operator both of which show a nontrivial dependence on the number of quarks flavours at the leading order in 1/N.     

An integrable noncommutative version of the sine-Gordon system

Using the bicomplex approach we discuss an integrable noncommutative system in two-dimensional Euclidean space. It is described by an equation of motion which reduces to the ordinary sine-Gordon equation when the noncommutation parameter is removed, plus a constraint equation which is nontrivial only in the noncommutative case. The implications of this constraint, which is required by integrability but seems to reduce the space of classical solutions, remain to be understood. We show that the system has an infinite number of conserved currents and we give the general recursive relation for constructing them. For the particular cases of lower spin nontrivial currents we work out the explicit expressions and perform a direct check of their conservation. These currents reduce to the usual sine-Gordon currents in the commutative limit. We find classical “localized” solutions to first order in the noncommutativity parameter and describe the Backlund transformations for our system. Finally, we comment on the relation of our noncommutative system to the commutative sine-Gordon system.     

Parton showers in leptoproduction events

We present a model that includes the production of arbitrarily many jets in lepton-hadron events, using the leading log formalism for parton shower evolution. The main problem encountered here, which has not previously been illuminated by studies ofe ⁺ e ^? annihilation or Drell-Yan/Z ⁰/W ^± production, is the choice of kinematics in the space-like shower evolution. In our preferred solution, the standard definition of Bjorkenx is preserved during the construction of initial and final state showers—a nontrivial constraint. The resulting model is described in detail, including some first investigations of its properties.     

Finite Size XXZ Spin Chain with Anisotropy Parameter Δ= \frac{1} {2}

We find an analytic solution of the Bethe–Ansatz equations (BAE) for the special case of a finite XXZ spin chain with free boundary conditions and with a complex surface field which provides for U_q(sl(2)) symmetry of the Hamiltonian. More precisely, we find one nontrivial solution, corresponding to the ground state of the system with anisotropy parameter Δ= $\frac{1}{2}$ corresponding to q³=?1. With a view to establishing an exact representation of the ground state of the finite size XXZ spin chain in terms of elementary functions, we concentrate on the crossing parameter η dependence around η=π/3 for which there is a known solution. The approach taken involves the use of a physical solution Q of Baxter's T-Q equation, corresponding to the ground state, as well as a non-physical solution P of the same equation. The calculation of P and then of the ground state derivative is covered. Possible applications of this derivative to the theory of percolation have yet to be investigated. As far as the finite XXZ spin chain with periodic boundary conditions is concerned, we find a similar solution for an assymetric case which corresponds to the 6-vertex model with a special magnetic field. For this case we find the analytic value of the “magnetic moment” of the system in the corresponding state.     

Novel scaling behavior of the Ising model on curved surfaces

We demonstrate the nontrivial scaling behavior of Ising models defined on (i) a donut-shaped surface and (ii) a curved surface with a constant negative curvature. By performing Monte Carlo simulations, we find that the former model has two distinct critical temperatures at which both the specific heat C(T) and magnetic susceptibility χ(T) show sharp peaks. The critical exponents associated with the two critical temperatures are evaluated by the finite-size scaling analysis; the result reveals that the values of these exponents vary depending on the temperature range under consideration. In the case of the latter model, it is found that static and dynamic critical exponents deviate from those of the Ising model on a flat plane; this is a direct consequence of the constant negative curvature of the underlying surface.     

Multiquark states

The pentaquark state recently discovered has been discussed based on various quark model calculations. Odd parity for the state cannot be ruled out theoretically because the contributions related to nontrivial color structures have not been studied completely. Other multiquark states, especially dibaryons, have been discussed also. A strangeness ?3 NΩ dibaryon has been shown to have a width as small as 12—22 keV and should be detectable in Ω-high-productivity reactions such as at RHIC, COMPAS, and the planned JHF and FAIR projects.     

Entropy increase for a class of dynamical maps

The dynamical evolution of a quantum system is described by a one parameter family of linear transformations of the space of self-adjoint trace class operators (on the Hilbert space of the system) into itself, which map statistical operators to statistical operators. We call such transformations dynamical maps. We give a sufficient condition for a dynamical map A not to decrease the entropy of a statistical operator. In the special case of an N-level system, this condition is also necessary and it is equivalent to the property that A preserves the central state.     

Dissipation-induced topological phase transition and periodic-driving-induced photonic topological state transfer in a small optomechanical lattice

We propose a scheme to investigate the topological phase transition and the topological state transfer based on the small optomechanical lattice under the realistic parameters regime.We find that the optomechanical lattice can be equivalent to a topologically nontrivial Su-Schrieffer Heeger(SSH)model via designing the effective optomechanical coupling.Especially,the optomechanical lattice experiences the phase transition between topologically nontrivial SSH phase and topologically trivial SSH phase by controlling the decay of the cavity field and the opto mechanical coupling.We stress that the to pological phase transition is mainly induced by the decay of the cavity field,which is counter-intuitive since the dissipation is usually detrimental to the system.Also,we investigate the photonic state transfer between the two cavity fields via the topologically protected edge channel based on the small optomechanical lattice.We find that the quantum st ate transfer assisted by the topological zero energy mode can be achieved via implying the external lasers with the periodical driving amplitudes into the cavity fields.Our scheme provides the fundamental and the insightful explanations towards the mapping of the photonic topological insulator based on the micro-nano optomechanical quantum optical platform.     

Skein theory and topological quantum registers: Braiding matrices and topological entanglement entropy of non-Abelian quantum Hall states

We study topological properties of quasi-particle states in the non-Abelian quantum Hall states. We apply a skein-theoretic method to the Read-Rezayi state whose effective theory is the SU(2)_K Chern-Simons theory. As a generalization of the Pfaffian (K = 2) and the Fibonacci (K = 3) anyon states, we compute the braiding matrices of quasi-particle states with arbitrary spins. Furthermore we propose a method to compute the entanglement entropy skein-theoretically. We find that the entanglement entropy has a nontrivial contribution called the topological entanglement entropy which depends on the quantum dimension of non-Abelian quasi-particle intertwining two subsystems.     

Coherent and Semiclassical States of a Charged Particle in Electromagnetic Fields

In the present article, we extend our study (Bagrov et al., Braz. J. Phys. 45, 369, 2015) of generalized coherent states (GCS) of a one-dimensional particle considering such important physical system as a three-dimensional charged particle in electric and magnetic fields. Constructing GCS in a many-dimensional case, we meet technical complications that make the consideration nontrivial and instructive. The GCS of the system under consideration are constructed. We study the properties of this GCS such as completeness relations, minimization of uncertainty relations, and so on. We point out which family of the obtained GCS of a charged particle in a magnetic field is related to the CS constructed first by Malkin and Man’ko. We obtain conditions under which some of the GCS can be considered as semiclassical states (SS).     

Dissipative dynamics of an extended magnetic nanostructure: spin necklace in a metallic environment

We study theoretically the dynamics of an "xxz" spin necklace coupled to a conduction electron sea, a model system for a nanostructure in a dissipative environment. We extract the long-time behavior via a mapping to a multichannel Coulomb gas problem followed by a scaling analysis. The strong quantum fluctuations of the necklace cause a nontrivial dependence of couplings on system size which we extract via an analysis involving the "boundary condition changing operator," and confirm via a detailed numerical evaluation of one case.     

Liquid–gas and other unusual thermal phase transitions in some large-N magnets

Much insight into the low temperature properties of quantum magnets has been gained by generalizing them to symmetry groups of order N, and then studying the large-N limit. In this paper we consider an unusual aspect of their finite temperature behavior—their exhibiting a phase transition between a perfectly paramagnetic state and a paramagnetic state with a finite correlation length at N=∞. We analyze this phenomenon in some detail in the large “spin” (classical) limit of the SU(N) ferromagnet which is also a lattice discretization of the CP^N−1 model. We show that at N=∞ the order of the transition is governed by lattice connectivity. At finite values of N, the transition goes away in one or less dimension but survives on many lattices in two dimensions and higher, for sufficiently large N. The latter conclusion contradicts a recent conjecture of Sokal and Starinets [Nucl. Phys. B 601 (2001) 425], yet is consistent with the known finite temperature behavior of the SU(2) case. We also report closely related first order paramagnet–ferromagnet transitions at large N and shed light on a violation of Elitzur's theorem at infinite N via the large-q limit of the q state Potts model, reformulated as an Ising gauge theory.     

On stability of solutions of theU(N) chiral model in two dimensions

We discuss the stability properties of classical solutions of theU(N) sigma models in two Euclidean dimensions. We show that all nontrivial solutions are unstable. For a general case we exhibit one mode of instability; in some special cases (corresponding to a grassmannian solution and an instantonic grassannian embedding) we exhibit two such independent modes.     

Unitary transformations purely based on geometric phases of two spins with Ising interaction in a rotating magnetic field

We consider how to obtain a nontrivial two-qubit unitary transformation purely based on geometric phases of two spin-'s with Ising-like interaction in a magnetic field with a static z-component and a rotating xy-component. This is an interesting problem both for the purpose of measuring the geometric phases and in quantum computing applications. In previous approach, coupling of one of the qubit with the rotating component of field is ignored. By considering the exact two-spin geometric phases, we find that a nontrivial two-spin unitary transformation purely based on Berry phases can be obtained by using two consecutive cycles with opposite directions of the magnetic field and opposite signs of the interaction constant. In the nonadiabatic case, starting with a certain initial state, a cycle in the projected space of rays and thus Aharonov-Anandan phase can be achieved. The two-cycle scheme cancels the total phases, hence any unknown initial state evolves back to itself without a phase factor.     

Perturbation theory of the Fermi surface in a quantum liquid. A general quasiparticle formalism and one-dimensional systems

We develop a perturbation theory formalism for the theory of the Fermi surface in a Fermi liquid of particles interacting via a bounded short-range repulsive pair potential. The formalism is based on the renormalization group and provides a formal expansion of the large-distance Schwinger functions in terms of a family of running couplings consisting of one- and two-body quasiparticle potentials. The flow of the running couplings is described in terms of a beta function, which is studied to all orders of perturbation theory and shown to obey, in thenth order,n! bounds. The flow equations are written in general dimensiond1 for the spinless case (for simplicity). The picture that emerges is that on a large scale the system looks like a system of fermions interacting via a-like interaction potential (i.e., a potential approaching 0 everywhere except at the origin, where it diverges, although keeping the integral bounded); the theory is not asymptotically free in the usual sense and the freedom mechanism is thus more delicate than usual: the technical problem of dealing with unbounded effective potentials is solved by introducing a mathematically precise notion ofquasiparticles, which turn out to be natural objects with finite interaction even when the physical potential diverges as a deltalike function. A remarkable kind of gauge symmetry is associated with the quasiparticles. To substantiate the analogy with the quasiparticle theory we discuss the mean field theory using our notion of quasiparticles: the resulting self-consistency relations are closely reminiscent of those of the BCS model. The formalism seems suited for a joint theory of normal states of Fermi liquids and of BCS states: the first are associated with the trivial fixed point of our flow or with nearby nontrivial fixed points (or invariant sets) and the second may naturally correspond to really nontrivial fixed points (which may nevertheless turn out to be accessible to analysis because the BCS state is a quasi free state, hence quite simple, unlike the nontrivial fixed points of field theory). Thed=1 case is deeply different from thed> 1 case, for our spinless fermions: we can treat it essentially completely for small coupling. The system is not asymptotically free and presents anomalous renormalization group flow with a vanishing beta function, and the discontinuity of the occupation number at the Fermi surface is smoothed by the interaction (remaining singular with a coupling-dependent singularity of power type with exponent identified with the anomalous dimension). Finally, we present a heuristic discussion of the theory for the flow of the running coupling constants in spinlessd> 1 systems: their structure is simplified further and the relevant part of the running interaction is precisely the interaction between pairs of quasiparticles which we identify with the Cooper pairs of superconductivity. The formal perturbation theory seems to have a chance to work only if the interaction between the Cooper pairs is repulsive: and to second order we show that in the spin-0 case this happens if the physical potential is repulsive. Our results indicate the possibility of the existence of a normal Fermi surface only if the interaction is repulsive.     

Conservation laws of generalized billiards that are polynomial in momenta

This paper deals with dynamics particles moving on a Euclidean n-dimensional torus or in an n-dimensional parallelepiped box in a force field whose potential is proportional to the characteristic function of the region D with a regular boundary. After reaching this region, the trajectory of the particle is refracted according to the law which resembles the Snell -Descartes law from geometrical optics. When the energies are small, the particle does not reach the region D and elastically bounces off its boundary. In this case, we obtain a dynamical system of billiard type (which was intensely studied with respect to strictly convex regions). In addition, the paper discusses the problem of the existence of nontrivial first integrals that are polynomials in momenta with summable coefficients and are functionally independent with the energy integral. Conditions for the geometry of the boundary of the region D under which the problem does not admit nontrivial polynomial first integrals are found. Examples of nonconvex regions are given; for these regions the corresponding dynamical system is obviously nonergodic for fixed energy values (including small ones), however, it does not admit polynomial conservation laws independent of the energy integral.     

Checking for optimal solutions in some NP-complete problems

For some weighted NP-complete problems, checking whether a proposed solution is optimal is a nontrivial task. Such is the case for the traveling salesman problem, or the spin-glass problem in three dimensions. In this Letter, we consider the weighted tripartite matching problem, a well known NP-complete problem. We write mean-field finite temperature equations for this model and derive their zero temperature limit. We show that any solution of the zero temperature equations provides an exact absolute ground state of the system. As a consequence, we propose a criterion which can be checked in polynomial time, and such that given a putative optimal solution, if the criterion is satisfied, then the solution is indeed optimal. This criterion is generalized to a class of variants of the multiple traveling salesmen problems.