Invariant Operators Research Articles

Local versus translationally invariant slowest operators in quantum Ising spin chains.

In this paper we study one-dimensional quantum Ising spin chains in an external magnetic field close to an integrable point. We concentrate on the dynamics of the slowest operator that plays a key role at the final period of thermalization. We introduce two independent definitions of the slowest operator: local and translationally invariant ones. We construct both operators numerically using tensor networks and extensively compare their physical properties. We find that the local operator has a significant overlap with energy flux, it does not correspond to an integral of motion, and, as one goes away from the integrable point, its revivals get suppressed and the rate of delocalization changes from extremely slow to slower than diffusion. The translationally invariant operator corresponds to an integral of motion; as the system becomes less integrable, at some point this operator changes its nature: from no overlap with any magnetization and fast rate of delocalization, to nonzero overlap with magnetizations σ_{x} and σ_{z} and slow rate of delocalization.

Generalized Willmore energies, Q-curvatures, extrinsic Paneitz operators, and extrinsic Laplacian powers

Over forty years ago, Paneitz, and independently Fradkin and Tseytlin, discovered a fourth-order conformally invariant differential operator, intrinsically defined on a conformal manifold, mapping scalars to scalars. This operator is a special case of the so-termed extrinsic Paneitz operator defined in the case when the conformal manifold is itself a conformally embedded hypersurface. In particular, this encodes the obstruction to smoothly solving the five-dimensional scalar Laplace equation, and suitable higher dimensional analogs, on conformally compact structures with constant scalar curvature. Moreover, the extrinsic Paneitz operator can act on tensors of general type by dint of being defined on tractor bundles. Motivated by a host of applications, we explicitly compute the extrinsic Paneitz operator. We apply this formula to obtain: an extrinsically-coupled [Formula: see text]-curvature for embedded four-manifolds, the anomaly in renormalized volumes for conformally compact five-manifolds with negative constant scalar curvature, Willmore energies for embedded four-manifolds, the local obstruction to smoothly solving the five-dimensional singular Yamabe problem, and new extrinsically-coupled fourth- and sixth-order operators for embedded surfaces and four-manifolds, respectively.

Noninvertible anomalies in SU(N) × U(1) gauge theories

We study 4-dimensional SU(N) × U(1) gauge theories with a single massless Dirac fermion in the 2-index symmetric/antisymmetric representations and show that they are endowed with a noninvertible 0-form {overset{sim }{mathbb{Z}}}_{2left(Npm 2right)}^{upchi} chiral symmetry along with a 1-form {mathbb{Z}}_N^{(1)} center symmetry. By using the Hamiltonian formalism and putting the theory on a spatial three-torus mathbbm{T} 3, we construct the non-unitary gauge invariant operator corresponding to {overset{sim }{mathbb{Z}}}_{2left(Npm 2right)}^{upchi} and find that it acts nontrivially in sectors of the Hilbert space characterized by selected magnetic fluxes. When we subject mathbbm{T} 3 to {mathbb{Z}}_N^{(1)} twists, for N even, in selected magnetic flux sectors, the algebra of {overset{sim }{mathbb{Z}}}_{2left(Npm 2right)}^{upchi} and {mathbb{Z}}_N^{(1)} fails to commute by a ℤ2 phase. We interpret this noncommutativity as a mixed anomaly between the noninvertible and the 1-form symmetries. The anomaly implies that all states in the torus Hilbert space with the selected magnetic fluxes exhibit a two-fold degeneracy for arbitrary mathbbm{T} 3 size. The degenerate states are labeled by discrete electric fluxes and are characterized by nonzero expectation values of condensates. In an appendix, we also discuss how to construct the corresponding noninvertible defect via the “half-space gauging” of a discrete one-form magnetic symmetry.

Automorphisms and strongly invariant relations

We investigate characterizations of the Galois connection {{,textrm{Aut},}}-{{,textrm{sInv},}} between sets of finitary relations on a base set A and their automorphisms. In particular, for A=omega _1, we construct a countable set R of relations that is closed under all invariant operations on relations and under arbitrary intersections, but is not closed under {textrm{sInv Aut}}. Our structure (A, R) has an omega -categorical first order theory. A higher order definable well-order makes it rigid, but any reduct to a finite language is homogeneous.

Shift Symmetries and AdS/CFT

Massive fields on anti-de Sitter (AdS) space enjoy galileon-like shift symmetries at particular values of their masses. We explore how these shift symmetries are realized through the boundary conformal field theory (CFT), at the level of the 2-point functions. In the alternate quantization scheme in which the dual conformal field gets the smaller ∆− conformal dimension, the shift symmetry is realized as a gauge symmetry in the dual CFT, so that only shift invariant operators are true conformal primary fields. In the standard quantization scheme the shift symmetry acts on the source, leading to Ward identities that take the form of integral constraints.

Maximal sectorial operators and invariant operator ranges

For unbounded maximal sectorial operators we establish necessary and sufficient conditions for the domain equality domA=domA⁎ and for the equality ReA=AR of operator real part ReA and form real part AR. Here ReA=12(A+A⁎) is half of the operator sum defined on domA∩domA⁎, whereas AR=12(A+˙A⁎) is the self-adjoint operator given by half of the form-sum of A and A⁎ so that, in general, ReA⊆AR. The natural question posed in [6], whether for a maximal sectorial operator A the equalitydomA=domA⁎implies the equalityReA=AR, is answered negatively in this paper. We construct families of unbounded coercive m-sectorial operators A such that domA=domA⁎ for which ReA is a closed symmetric non-selfadjoint operator or a non-closed essentially selfadjoint operator. Moreover, we show that the domain equalities domA=domA⁎ and domReA=domAR are equivalent to problems of invariant operator ranges of bounded selfadjoint or unitary operators as well as to the existence of bounded operators with specific operator range properties.

Einstein Manifold of Pseudo-Orthogonal Group and Invariant Differential Operators

Einstein Manifold of Pseudo-Orthogonal Group and Invariant Differential Operators

Werner states from diagrams

We present two results on multiqubit Werner states, defined to be those states that are invariant under the collective action of any given single-qubit unitary that acts simultaneously on all the qubits. Motivated by the desire to characterize entanglement properties of Werner states, we construct a basis for the real linear vector space of Werner invariant Hermitian operators on the Hilbert space of pure states; it follows that any mixed Werner state can be written as a mixture of these basis operators with unique coefficients. Continuing a study of ‘polygon diagram’ Werner states constructed in earlier work, with a goal to connect diagrams to entanglement properties, we consider a family of multiqubit states that generalize the singlet, and show that their 2-qubit reduced density matrices are separable.

Anomalous dimensions from soft Regge constants

Using an effective field theory (EFT) formalism for forward scattering, we reconsider the factorization of 2 → 2 scattering amplitudes in the Regge limit. Expanding the amplitude in gauge invariant operators labelled by the number of Glauber exchanges, allows us to further factorize the standard impact factors into separate collinear and soft functions. The soft functions are universal, and describe radiative corrections to the Reggeized gluon states exchanged by the collinear projectiles. Remarkably, we find that the one-loop soft function for the single Reggeized gluon state is given to mathcal{O} (ϵ) in terms of the two-loop cusp and two-loop rapidity anomalous dimensions. We argue that this iterative structure follows from the simple action of crossing symmetry in the forward scattering limit, which in the EFT allows us to replace the divergent part of a soft loop by a much simpler Glauber loop. We use this correspondence to provide a simple calculation of the two-loop Regge trajectory using the EFT. We then explore its implications at higher perturbative orders, and derive the maximally matter dependent contributions to the Regge trajectory to all loop orders, i.e. the terms sim {alpha}_s^{k+1}{n}_f^k for any k, where nf is the number of massless flavors. These simplifications suggests that the EFT approach to the Regge limit will be helpful to explore and further understand the structure of the Regge limit.

Complexity in the Lipkin-Meshkov-Glick model.

We study complexity in a spin system with infinite-range interaction, via the paradigmatic Lipkin-Meshkov-Glick (LMG) model, in the thermodynamic limit. Exact expressions for the Nielsen complexity (NC) and the Fubini-Study complexity (FSC) are derived, which helps us to establish several distinguishing features compared to complexity in other known spin models. In a time-independent LMG model, close to phase transition, the NC diverges logarithmically, much like the entanglement entropy. Remarkably, however, in a time-dependent scenario, this divergence is replaced by a finite discontinuity, as we show by using the Lewis-Riesenfeld theory of time-dependent invariant operators. The FSC of a variant of the LMG model shows different behavior compared to quasifree spin models. Namely, it diverges logarithmically when the target (or reference) state is near the separatrix. Numerical analysis indicates that this is due to the fact that geodesics starting with arbitrary boundary conditions are "attracted" toward the separatrix and that near this line, a finite change in the affine parameter of the geodesic produces an infinitesimal change of the geodesic length. The same divergence is shared by the NC of this model as well.

Decoherence and Transition to Classicality for Time-Dependent Stochastic Quantum Systems with a General Environment

The emergence of classicality from a stochastic quantum system through decoherence is investigated. We consider the case where the parameters, such as mass, frequency, and the damping coefficient, vary with time. The invariant operator theory is employed in order to describe quantum evolution of the system. It is supposed that the system is in equilibrium with the environment at a finite temperature. The characteristics of decoherence, the classical correlation and the quantum coherence length are analyzed. The decoherence time is estimated in both position and momentum spaces. We verify from such analyses that the time dependence of the stochastic process affects the quantum-to-classical transition of the system. To promote the understanding of the results, we apply our development to a particular system which is the damped harmonic oscillator. Through this application, we confirm that the decoherence condition is satisfied in the limit of a sufficiently high temperature, whereas the classical correlation is not affected by the temperature.

On the two-dimensional time-dependent anisotropic harmonic oscillator in a magnetic field

A charged harmonic oscillator in a magnetic field, Landau problems, and an oscillator in a noncommutative space share the same mathematical structure in their Hamiltonians. We have considered a two-dimensional anisotropic harmonic oscillator with arbitrarily time-dependent parameters (effective mass and frequencies), placed in an arbitrarily time-dependent magnetic field. A class of quadratic invariant operators (in the sense of Lewis and Riesenfeld) have been constructed. The invariant operators (Î) have been reduced to a simplified representative form by a linear canonical transformation [the group Sp(4,R)]. An orthonormal basis of the Hilbert space consisting of the eigenvectors of Î is obtained. In order to obtain the solutions of the time-dependent Schrödinger equation corresponding to the system, both the geometric and dynamical phase-factors are constructed. A generalized Peres–Horodecki separability criterion (Simon’s criterion) for the ground state corresponding to our system has been demonstrated.

Real hypersurfaces with Reeb invariant shape operator in the complex hyperbolic quadric

Real hypersurfaces with Reeb invariant shape operator in the complex hyperbolic quadric

Curved Versions of the Ovsienko–Redou Operators

Abstract We construct a family of conformally covariant bidifferential operators on pseudo-Riemannian manifolds. Our construction is analogous to the construction of Graham–Jenne–Mason–Sparling of conformally covariant differential operators via tangential powers of the Laplacian in the Fefferman–Graham ambient space. In fact, we completely classify the tangential bidifferential operators on the ambient space, which are expressed purely in terms of the ambient Laplacian. This gives a curved analogue of the classification, due to Ovsienko–Redou and Clerc, of conformally invariant bidifferential operators on the sphere. As an application, we construct a large class of formally self-adjoint conformally invariant differential operators.

General Toeplitz kernels and -invariance

AbstractMotivated by the near invariance of model spaces for the backward shift, we introduce a general notion of $(X,Y)$ -invariant operators. The relations between this class of operators and the near invariance properties of their kernels are studied. Those lead to orthogonal decompositions for the kernels, which generalize well-known orthogonal decompositions of model spaces. Necessary and sufficient conditions for those kernels to be nearly X-invariant are established. This general approach can be applied to a wide class of operators defined as compressions of multiplication operators, in particular to Toeplitz operators and truncated Toeplitz operators, to study the invariance properties of their kernels (general Toeplitz kernels).

Circuit Complexity in Interacting Quenched Quantum Field Theory

In this work, we explore the effects of quantum quenching on the circuit complexity of quenched quantum field theory with weakly coupled quartic interactions. We use the invariant operator method under a perturbative framework to compute the ground state of this system. We give the analytical expressions for specific reference and target states using the ground state of the system. Using a particular cost functional, we show the analytical computation of circuit complexity for the quenched and interacting field theory. Furthermore, we give a numerical estimate of circuit complexity with respect to the quench rate, δt, for two coupled oscillators. The parametric variation in the unambiguous contribution of the circuit complexity for an arbitrary number of oscillators has been studied with respect to the dimensionless parameter (t/δt). We comment on the variation in the circuit complexity for different values of coupling strength, different numbers of oscillators and even in different dimensions.

Heat- and Wave-Type Equations with Nonlocal Operators, I. Compact Lie Groups

Abstract We prove existence and uniqueness and give the analytical solution of heat and wave type equations on a compact Lie group $G$ by using a nonlocal (in time) differential operator and a positive left invariant operator (maybe unbounded) acting on the group. For heat type equations, solutions are given in $L^q(G)$ for data in $L^p(G)$ with $1<p\leqslant 2\leqslant q<+\infty $. We also provide some asymptotic estimates (large-time behavior) for the solutions. Some examples are given. Also, for wave-type equations, we give the solution on some suitable Sobolev spaces over $L^2(G)$. We complement our results, by studying a multi-term heat-type equation as well.

The Fourth-Linear aff(1)-Invariant Differential Operators and the First Cohomology of the Lie Algebra of Vector Fields on RP1

In this paper, we denote the Lie algebra of smooth vector fields on RP1 by V(RP1). This paper focuses on two parts. In the beginning, we determine the cohomology space of aff(1) with coefficients in Dτ,λ,μ;ν. Afterward, we classify aff(1)-invariant fourth-linear differential operators from V(RP1) to Dτ,λ,μ;ν vanishing on aff(1). This result enables us to compute the aff(1)-relative cohomology of V(RP1) with coefficients in Dτ,λ,μ;ν.

On the quantum dynamics of a general time-dependent coupled oscillator

By using the Lewis–Riesenfeld invariants theory, we investigate the quantum dynamics of a two-dimensional (2D) time-dependent coupled oscillator. We introduce a unitary transformation and show the conditions under which the invariant operator is uncoupled to describe two simple harmonic oscillators with time-independent frequencies and unit masses. The decouplement allows us to easily obtain the corresponding eigenstates. The generalization to three-dimensional (3D) time-dependent coupled oscillator is briefly discussed where a diagonalized invariant, which is exactly the sum of three simple harmonic oscillators, is obtained under specific conditions on the parameters.

Invariants of formal pseudodifferential operator algebras and algebraic modular forms

We study from an algebraic point of view the question of extending an action of a group Γ on a commutative domain R to a formal pseudodifferential operator ring B = R((x ; d)) with coefficients in R, as well as to some canonical quadratic extension C = R((x 1/2 ; 1 2 d))2 of B. We give a necessary and sufficient condition of compatibility between the action and the derivation d of R for such an extension to exist, and we determine all possible extensions of the action to B and C. We describe under suitable assumptions the invariant subalgebras B Γ and C Γ as Laurent series rings with coefficients in R Γ. The main results of this general study are applied in a numbertheoretical context to the case where Γ is a subgroup of SL(2, C) acting by homographies on an algebra R of functions in one complex variable. Denoting by Mj the vector space of algebraic modular forms in R of weight j (even or odd), we build for any nonnegative integer k a linear isomorphism between the subspace C Γ k of invariant operators of order ≥ k in C Γ and the product space M k = j≥k Mj, which can be identified with a space of algebraic Jacobi forms of weight k. It results in particular a structure of noncommutative algebra on M0 and an algebra isomorphism Ψ : M0 → C Γ 0 , whose restriction to the particular case of even weights was previously known in the litterature. We study properties of this correspondence combining arithmetical arguments and the use of the algebraic results of the first part of the article.