Nonlocal Transformation Research Articles

From symmetry-protected topological order to Landau order

Focusing on the particular case of the discrete symmetry group Z_N x Z_N, we establish a mapping between symmetry protected topological phases and symmetry broken phases for one-dimensional spin systems. It is realized in terms of a non-local unitary transformation which preserves the locality of the Hamiltonian. We derive the image of the mapping for various phases involved, including those with a mixture of symmetry breaking and topological protection. Our analysis also applies to topological phases in spin systems with arbitrary continuous symmetries of unitary, orthogonal and symplectic type. This is achieved by identifying suitable subgroups Z_N x Z_N in all these groups, together with a bijection between the individual classes of projective representations.

Dynamics of geometric phase for composite quantum system under nonlocal unitary transformation: a case study of dimer system

In this paper, we explore the dynamical properties of geometric phase for a composite quantum system under the nonlocal unitary evolution. As an illustrative example, the analytical expressions of geometric phase are derived for the dimer system. We find that geometric phase presents some interesting properties with coupling strengths (corresponding to nonlocal unitary evolution), such as dynamical oscillation behavior with time evolution, monotonicity, symmetry, etc. We show that the geometric phase and entanglement have the same period for some conditions. Moreover, we discuss geometric phase of the whole system and its subsystems. Our investigations show that geometric phase can reflect some inherent properties of the system: it signals a transition from self-trapping to delocalization.

Schrödinger sigma models and Jordanian twists

We proceed to study the integrable structures of two-dimensional non-linear sigma models defined on three-dimensional Schrodinger spacetimes. We show that anisotropic Lax pairs are equivalent with isotropic Lax pairs with flat conserved currents under non-local gauge transformations. Then a quite non-trivial realization of the undeformed Yangian symmetry Y(sl(2)) is revealed after an appropriate gauge fixing, which is determined by comparing the gauge transformation to a quantum Jordanian twist. As a result, an exotic symmetry found in arXiv:1209.4147 may be interpreted as a Jordanian deformation of Y(sl(2)).

Exactly solvable two-dimensional stationary Schrödinger operators obtained by the nonlocal Darboux transformation

The Fokker-Planck equation associated with the two - dimensional stationary Schr\"odinger equation has the conservation low form that yields a pair of potential equations. The special form of Darboux transformation of the potential equations system is considered. As the potential variable is a nonlocal variable for the Schr\"odinger equation that provides the nonlocal Darboux transformation for the Schr\"odinger equation. This nonlocal transformation is applied for obtaining of the exactly solvable two - dimensional stationary Schr\"odinger equations. The examples of exactly solvable two - dimensional stationary Schr\"odinger operators with smooth potentials decaying at infinity are obtained.

Generating finite dimensional integrable nonlinear dynamical systems

In this article, we present a brief overview of some of the recent progress made in identifying and generating finite dimensional integrable nonlinear dynamical systems, exhibiting interesting oscillatory and other solution properties, including quantum aspects. Particularly we concentrate on Lienard type nonlinear oscillators and their generalizations and coupled versions. Specific systems include Mathews-Lakshmanan oscillators, modified Emden equations, isochronous oscillators and generalizations. Nonstandard Lagrangian and Hamiltonian formulations of some of these systems are also briefly touched upon. Nonlocal transformations and linearization aspects are also discussed.

Exact spin-orbital separation in a solvable model in one dimension

A one-dimensional model of coupled spin-1/2 spins and pseudospin-1/2 orbitals with nearest-neighbor interaction is rigorously shown to exhibit spin-orbital separation by means of a non-local unitary transformation. On an open chain, this transformation completely decouples the spins from the orbitals in such a way that the spins become paramagnetic while the orbitals form the soluble XXZ Heisenberg model. The nature of various correlations is discussed. The more general cases, which allow spin-orbital separation by the same method, are pointed out. A generalization for the orbital pseudospin greater than 1/2 is also discussed. Some qualitative connections are drawn with the recently observed spin-orbital separation in Sr2CuO3.

Uniqueness of the Fock quantization of scalar fields under mode preserving canonical transformations varying in time

We study the Fock quantization of scalar fields of Klein-Gordon type in nonstationary scenarios propagating in spacetimes with compact spatial sections, allowing for different field descriptions that are related by means of certain nonlocal linear canonical transformations that depend on time. More specifically, we consider transformations that do not mix eigenmodes of the Laplace-Beltrami operator, which are supposed to be dynamically decoupled. In addition, we assume that the canonical transformations admit an asymptotic expansion for large eigenvalues (in norm) of the Laplace-Beltrami operator in the form of a series of half integer powers. Canonical transformations of this kind are found in the study of scalar perturbations in inflationary cosmologies, relating for instance the physical degrees of freedom of these perturbations after gauge fixing with gauge invariant canonical pairs of Bardeen quantities. We characterize all possible transformations of this type and show that, independently of the initial field description, the combined criterion of requiring (i) invariance of the vacuum under the spatial symmetries and (ii) a unitary implementation of the dynamics, leads to a unique equivalence class of Fock quantizations, all of them related by unitary transformations. This conclusion provides even further robustness to the validity of the proposed criterion, completing the results that have already appeared in the literature about the uniqueness of the Fock quantization under changes of field description when one permits exclusively local time dependent canonical transformations that scale the field configuration.

A Short Note on Nonlocal Transformations and First Integrals for Certain Nonlinear Oscillator Equations

This short note presents a simple and effective application of the generalized Sundman transformations and the Jacobi last multiplier approach for two nonlinear oscillator equations. We reobtain the first integrals with this method very easily. In addition, we also compute the Sundman symmetries and Lagrangians.

Zel’dovich approximation and general relativity

Abstract We show how the Zel’dovich approximation and the second-order displacement field of Lagrangian perturbation theory can be obtained from a general relativistic gradient expansion in Λ cold dark matter cosmology. The displacement field arises as a result of a second-order non-local coordinate transformation which brings the synchronous/comoving metric into a Newtonian form. We find that, with a small modification, the Zel’dovich approximation holds even on scales comparable to the horizon. The corresponding density perturbation is not related to the Newtonian potential via the usual Poisson equation but via a modified Helmholtz equation. This is a consequence of causality not present in the Newtonian theory. The second-order displacement field receives relativistic corrections that are subdominant on short scales but are comparable to the second-order Newtonian result on scales approaching the horizon. The corrections are easy to include when setting up initial conditions in large N-body simulations.

Three-point partial difference equations linearizable by local and nonlocal transformations

We consider a class of nonlinear partial difference equations defined on three points of a plane lattice. We construct conditions for this class of partial difference equations to be linearizable through a point or a Cole–Hopf transformation. Using these conditions we are able to classify all multilinear linearizable equations belonging to this class.

On non-multiaffine consistent-around-the-cube lattice equations

We show that integrable involutive maps, due to the fact they admit three integrals in separated form, can give rise to equations, which are consistent around the cube and which are not in the multiaffine form assumed in Adler et al. (2003) [1]. Lattice models, which are discussed here, are related to the lattice potential KdV equation by nonlocal transformations (discrete quadratures).

λ‐symmetries, nonlocal transformations and first integrals to a class of Painlevé–Gambier equations

In this work, we consider a class of Painlevé–Gambier equations that model the motion of chain ball drawing with constant force in the frictionless surface. λ‐symmetries, first integrals, integrating factors, nonlocal transformations and local transformations are derived by using the some recent studies that are proposed by Muriel and Romero. Copyright © 2012 John Wiley & Sons, Ltd.

Two-dimensional symmetry-protected topological orders and their protected gapless edge excitations

Topological insulators in free fermion systems have been well characterized and classified. However, it is not clear in strongly interacting boson or fermion systems what symmetry protected topological orders exist. In this paper, we present a model in a 2D interacting spin system with nontrivial on-site $Z_2$ symmetry protected topological order. The order is nontrivial because we can prove that the 1D system on the boundary must be gapless if the symmetry is not broken, which generalizes the gaplessness of Wess-Zumino-Witten model for Lie symmetry groups to any discrete symmetry groups. The construction of this model is related to a nontrivial 3-cocycle of the $Z_2$ group and can be generalized to any symmetry group. It potentially leads to a complete classification of symmetry protected topological orders in interacting boson and fermion systems of any dimension. Specifically, this exactly solvable model has a unique gapped ground state on any closed manifold and gapless excitations on the boundary if $Z_2$ symmetry is not broken. We prove the latter by developing the tool of matrix product unitary operator to study the nonlocal symmetry transformation on the boundary and revealing the nontrivial 3-cocycle structure of this transformation. Similar ideas are used to construct a 2D fermionic model with on-site $Z_2$ symmetry protected topological order.

Gauging away physics

We consider the recent argument by Higuchi et al(2011 Class. Quantum Grav. 28 245012) that a nonlocal gauge transformation can be used to eliminate the infrared divergence of the graviton propagator, when evaluated in Bunch–Davies vacuum on the open coordinate submanifold of de Sitter space in the transverse-traceless-synchronous gauge. Because the transformation is not local, the equal time commutator of undifferentiated fields no longer vanishes. From explicit examination of the Wightman function we demonstrate that the transformation adds anti-sources in the far future which cancel the bad infrared behavior but also change the propagator equation. The same problem exists in the localized version of the recent argument. Adding such anti-sources does not seem to be legitimate and could be used to eliminate the infrared divergence of the massless, minimally coupled scalar. The addition of such anti-sources in flat space QED could be effected by an almost identical gauge transformation, and would seem to eliminate the well-known infrared divergences which occur in loop corrections to exclusive amplitudes.

Thermo mechanical buckling analysis of carbon nanotubes on winkler foundation using non-local elasticity theory and DTM

Thermo-mechanical buckling behaviour of Single Walled Carbon Nanotube (SWCNT) on Winkler foundation is reported. Non-local elasticity theory and differential transformation method are applied. Effects of (i) size of SWCNT, (ii) non-local parameter, (iii) Winkler modulus, (iv) boundary conditions and (v) temperature on buckling load are investigated.

TENSOR MODELS AND HIERARCHY OF n-ARY ALGEBRAS

Tensor models are generalization of matrix models, and are studied as models of quantum gravity. It is shown that the symmetry of the rank-three tensor models is generated by a hierarchy of n-ary algebras starting from the usual commutator, and the 3-ary algebra symmetry reported in the previous paper is just a single sector of the whole structure. The condition for the Leibnitz rules of the n-ary algebras is discussed from the perspective of the invariance of the underlying algebra under the n-ary transformations. It is shown that the n-ary transformations which keep the underlying algebraic structure invariant form closed finite n-ary Lie subalgebras. It is also shown that, in physical settings, the 3-ary transformation practically generates only local infinitesimal symmetry transformations, and the other more nonlocal infinitesimal symmetry transformations of the tensor models are generated by higher n-ary transformations.

A λ-symmetry-based method for the linearization and determination of first integrals of a family of second-order ordinary differential equations

We present a characterization of the second-order ordinary differential equations (ODEs) that can be linearized by means of certain nonlocal transformations. This characterization is given in terms of the coefficients of the equation and also determines the second-order ODEs that admit λ-symmetries and first integrals of some specific forms. Systematic methods to find these first integrals, λ-symmetries and the linearizing transformations are also derived.

Nonlocal symmetries and formulae for generation of solutions for a class of diffusion–convection equations

New formulae of nonlocal nonlinear superposition and generation of solutions are proposed for nonlinear diffusion–convection equations which are linearizable or are invariant with respect to a generalized hodograph transformation or connected by this transformation. We study in what particular ways additional Lie symmetries of diffusion–convection equations induce nonlocal symmetries of equations obtained from the initial ones by nonlocal transformations. The formulae derived are used for the construction of exact solutions.

Topological disentangler for the valence-bond-solid chain

We discuss topological disentangler for S=1 quantum spin chains in the Haldane phase. We first point out that Kennedy-Tasaki's(KT) nonlocal unitary transformation is the perfect disentangler for Affleck-Kennedy-Lieb-Tasaki model. We then demonstrate that the KT transformation can be reconstructed as an assembly of pair disentanglers. Finally, we show that the KT transformation can be regarded as a topological disentangler, which selectively disentangles the double-fold degeneracy in the entanglement spectrum of the S=1 Heisenberg chain.

Nonlocal Symmetries of Systems of Evolution Equations

We prove that any potential symmetry of a system of evolution equations reduces to a Lie symmetry through a nonlocal transformation of variables. This fact is in the core of our approach to computation of potential and more general nonlocal symmetries of systems of evolution equations having nontrivial Lie symmetry. Several examples are considered.