Matrix Elements Of Operators Research Articles

Computing observables without eigenstates: Applications to Bloch Hamiltonians

Calculating the observables of a Hamiltonian requires taking matrix elements of operators in the eigenstate basis. Since eigenstates are only defined up to arbitrary phases that depend on Hamiltonian parameters, analytical expressions for observables are often difficult to simplify. In this work, we show how for small Hilbert space dimension N all observables can be expressed in terms of the Hamiltonian and its eigenvalues using the properties of the SU(N) algebra, and we derive explicit expressions for N=2,3,4. Then we present multiple applications specializing to the case of Bloch electrons in crystals, including the computation of Berry curvature, quantum metric and orbital moment, as well as a more complex observable in non-linear response, the linear photogalvanic effect (LPGE). As a physical example we consider multiband Hamiltonians with nodal degeneracies to show first how constraints between these observables are relaxed when going from two to three-band models, and second how quadratic dispersion can lead to constant LPGE at small frequencies.

Non-perturbative renormalization scheme for the C P -odd three-gluon operator

We define a regularization-independent momentum-subtraction scheme for the C P -odd three-gluon operator at dimension six. This operator appears in effective field theories for heavy physics beyond the Standard Model, describing the indirect effect of new sources of C P-violation at low energies. In a hadronic context, it induces permanent electric dipole moments. The hadronic matrix elements of the three-gluon operator are non-perturbative objects that should ideally be evaluated with lattice QCD. We define a non-perturbative renormalization scheme that can be implemented on the lattice and we compute the scheme transformation to overline{mathrm{MS}} at one loop. Our calculation can be used as an interface to future lattice-QCD calculations of the matrix elements of the three-gluon operator, in order to obtain theoretically robust constraints on physics beyond the Standard Model from measurements of the neutron electric dipole moment.

Regularization, renormalization and consistency conditions in QED with x-electric potential steps

The present article is an important addition to the nonperturbative formulation of QED with x-steps presented by Gavrilov and Gitman (Phys. Rev. D. 93:045002, 2016). Here we propose a new renormalization and volume regularization procedures which allow one to calculate and distinguish physical parts of different matrix elements of operators of the current and of the energy–momentum tensor, at the same time relating the latter quantities with characteristics of the vacuum instability. For this purpose, a modified inner product and a parameter tau of the regularization are introduced. The latter parameter can be fixed using physical considerations. In the Klein zone this parameter can be interpreted as the time of the observation of the pair-production effect. In the refined formulation of QED with x-steps, we succeeded to consider the back-reaction problem. In the case of an uniform electric field E confined between two capacitor plates separated by a finite distance L, we see that the smallness of the back-reaction implies a restriction (the consistency condition) on the product EL from above.

Short flow-time coefficients of CP -violating operators

Measurements of a permanent neutron electric dipole moment (EDM) potentially probe Beyond-the-Standard Model (BSM) sources of CP-violation. At low energy the CP-violating BSM interactions are parametrized by flavor-conserving CP-violating operators of dimension higher than four. QCD calculations of the nucleon matrix elements of these operators are required to fully reconstruct the sources and magnitudes of the different CP-violating contributions to the nucleon EDM. Herein we study the quark-chromo electric dipole moment (qCEDM) operator and the three-gluon Weinberg operator. The non-perturbative determination, using lattice QCD, of the nucleon matrix elements of these CP-violating operators is hampered by their short-distance behavior. Under renormalization these operators mix with lower dimensional operators, which induces power divergences in the lattice spacing, as the continuum limit is approached. We study the short-distance behavior of the qCEDM and the Weinberg operators using the gradient flow. We perform a short flow time expansion and determine, in perturbation theory, the expansion coefficients of the linearly-divergent terms stemming from the mixing with the pseudoscalar density and the topological charge, confirming the expectations of the operator product expansion. We introduce a new method to perform calculations at non-zero flow-time for arbitrary values of the external momenta. This method allows us to work in four dimensions for most of the calculations described in this paper, avoiding the complications associated with defining $\gamma_5$ in generic d dimensions. We show that leading contributions in the external momenta can be reproduced by defining $\gamma_5$ using the 't Hooft-Veltman-Breitenlohner-Maison scheme.

Semiclassical theory of long time propagation in quantum chaos. First part

We develop a semiclassical theory of wave propagation based on invariant Lagrangian manifolds existing in conservative Hamiltonian systems with chaotic dynamics. They are stable and unstable manifolds of unstable periodic orbits, and their intersections consist of homoclinic and heteroclinic orbits. For arbitrary long times, we find matrix elements of the evolution operator between wave functions constructed in the neighbourhood of short unstable periodic orbits, in terms of canonical invariants of homoclinic and heteroclinic orbits. We verify the accuracy of these expressions by computing millions of homoclinic orbits and thousands of heteroclinic ones in the hyperbola billiard. Here we describe diagonal matrix elements while in another article named second part we will describe off-diagonal matrix elements.

Matrix elements of vertex operators and fermionic limit of spin Calogero-Sutherland system

We present a construction of an integrable model as a projective type limit of spin Calogero-Sutherland model with N fermionic particles, where N tends to infinity. It is implemented in the multicomponent fermionic Fock space. Explicit formulas for limits of Dunkl operators and the Yangian generators are presented by means of fermionic fields.

Low-frequency behavior of off-diagonal matrix elements in the integrable XXZ chain and in a locally perturbed quantum-chaotic XXZ chain

We study the matrix elements of local operators in the eigenstates of the integrable XXZ chain and of the quantum-chaotic model obtained by locally perturbing the XXZ chain with a magnetic impurity. We show that, at frequencies that are polynomially small in the system size, the behavior of the variances of the off-diagonal matrix elements can be starkly different depending on the operator. In the integrable model we find that, as the frequency $\omega\rightarrow0$, the variances are either nonvanishing (generic behavior) or vanishing (for a special class of operators). In the quantum-chaotic model, on the other hand, we find the variances to be nonvanishing as $\omega\rightarrow0$ and to indicate diffusive dynamics. We highlight which properties of the matrix elements of local operators are different between the integrable and quantum-chaotic models independently of the specific operator selected.

Eigenstate Thermalization in a Locally Perturbed Integrable System.

Eigenstate thermalization is widely accepted as the mechanism behind thermalization in generic isolated quantum systems. Using the example of a single magnetic defect embedded in the integrable spin-1/2 XXZ chain, we show that locally perturbing an integrable system can give rise to eigenstate thermalization. Unique to such setups is the fact that thermodynamic and transport properties of the unperturbed integrable chain emerge in properties of the eigenstates of the perturbed (nonintegrable) one. Specifically, we show that the diagonal matrix elements of observables in the perturbed eigenstates follow the microcanonical predictions for the integrable model, and that the ballistic character of spin transport in the integrable model is manifest in the behavior of the off-diagonal matrix elements of the current operator in the perturbed eigenstates.

Inclusive probability of particle creation on classical backgrounds

The quantum theories of boson and fermion fields with quadratic nonstationary Hamiltoanians are rigorously constructed. The representation of the algebra of observables is given by the Hamiltonian diagonalization procedure. The sufficient conditions for the existence of unitary dynamics at finite times are formulated and the explicit formula for the matrix elements of the evolution operator is derived. In particular, this gives the well-defined expression for the one-loop effective action. The ultraviolet and infrared divergencies are regularized by the energy cutoff in the Hamiltonian of the theory. The possible infinite particle production is regulated by the corresponding counterdiabatic terms. The explicit formulas for the average number of particles N_D recorded by the detector and for the probability w(D) to record a particle by the detector are derived. It is proved that these quantities allow for no-regularization limit and, in this limit, N_D is finite and w(D)in [0,1). As an example, the theory of a neutral boson field with stationary quadratic part of the Hamiltonian and nonstationary source is considered. The average number of particles produced by this source from the vacuum during a finite time evolution and the inclusive probability to record a created particle are obtained. The infrared and ultraviolet asymptotics of the average density of created particles are derived. As a particular case, quantum electrodynamics with a classical current is considered. The ultraviolet and infrared asymptotics of the average number of photons are derived. The asymptotics of the average number of photons produced by the adiabatically driven current is found.

Manifestations of Quantum Anomalies of Field Theory in Quantum Statistical Mechanics

A new class of relations for statistically averaged matrix elements of different operators (such as Hamiltonian and conserved currents) is described in the one-loop approximation. The matrix elements have polynomial dependence on temperature and other thermodynamic values characterizing the equilibrium of the medium (the chemical potential, the angular velocity of rotation, and the acceleration). In this sense, the situation is analogous to the chiral anomaly in quantum field theory, which fixes the divergence of the axial current as a polynomial in external electromagnetic fields. Moreover, in this special case of matrix element of the axial current, it is possible to establish the relationship between the derivation of the anomaly and the one-loop expressions of statistical physics. The central role in establishing the correspondence is played by the polynomial Sommerfeld integrals. The generalizations of the one-loop relations in statistical physics are proposed, which (at least, today) have no analogs in quantum field theory.

Direct measurement methods of density matrix of an entangled quantum state

In general, the state of a quantum system represented by the density operator and its determination is a fundamental problem in quantum theory. In this study, two theoretical methods such as using postselected measurement characterized by modular value and sequential measurements of triple products of complementary observables to direct measurement of matrix elements of density operator of a two photon entangled quantum state are introduced. The similarity and possible feasibilities of those two methods are discussed by considering the previous theoretical and experimental works.

Strong integrability of the bi-YB–WZ model

We identify the r-matrix governing the Poisson brackets of the matrix elements of the Lax operator of the bi-YB-WZ model.

Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs

<p style='text-indent:20px;'>Starting from the matrix elements of a nucleon-nucleon potential operator provided in a basis of spherical harmonic oscillator functions, we present an algorithm for expressing a given potential operator in terms of irreducible tensors of the SU(3) and SU(2) groups. Further, we introduce a GPU-based implementation of the latter and investigate its performance compared with a CPU-based version of the same. We find that the CUDA implementation delivers speedups of 2.27x – 5.93x.

Scattering using real-time path integrals

Background: Path integrals are a powerful tool for solving problems in quantum theory that are not amenable to a treatment by perturbation theory. Most path integral computations require an analytic continuation to imaginary time. While imaginary time treatments of scattering are possible, imaginary time is not a natural framework for treating scattering problems. Purpose: To test a recently introduced method for performing direct calculations of scattering observables using real-time path integrals. Methods: The computations are based on a new interpretation of the path integral as the expectation value of a potential functional on a space of continuous paths with respect to a complex probability distribution. The method has the advantage that it can be applied to arbitrary short-range potentials. Results: The new method is tested by applying it to calculate half-shell sharp-momentum transition matrix elements for one-dimensional potential scattering. The calculations for half shell transition operator matrix elements are in agreement with a numerical solution of the Lippmann-Schwinger equation. The computational method has a straightforward generalization to more complicated systems.

Quantum thermalization and Virasoro symmetry

We initiate a systematic study of high energy matrix elements of local operators in 2D CFT. Knowledge of these is required in order to determine whether the generalized eigenstate thermalization hypothesis (ETH) can hold in such theories. Most high energy states are high level Virasoro descendants, and by employing an oscillator representation of the Virasoro algebra we develop an efficient method for computing matrix elements of primary operators in such states. In parameter regimes where we expect (e.g. from AdS/CFT intuition) thermalization to occur, we observe striking patterns in the matrix elements: diagonal matrix elements are smoothly varying and off-diagonal elements, while nonzero, are power-law suppressed compared to the diagonal elements. We discuss the implications of these universal properties of 2D CFTs in regard to their compatibility with generalized ETH.

Finite-size effects in cylindrical topological insulators

We present a theoretical study of a nanowire made of a three-dimensional topological insulator. The bulk topological insulator is described by a continuum-model Hamiltonian, and the cylindrical-nanowire geometry is modelled by a hard-wall boundary condition. We provide the secular equation for the eigenergies of the systems (both for bulk and surface states) and the analytical form of the energy eigenfunctions. We describe how the surface states of the cylinder are modified by finite-size effects. In particular, we provide a 1/R expansion for the energy of the surface states up to second order. The knowledge of the analytical form for the wavefunctions enables the computation of matrix elements of any single-particle operators. In particular, we compute the matrix elements of the optical dipole operator, which describe optical absorption and emission, treating intra- and inter-band transition on the same footing. Selection rules for optical transitions require conservation of linear momentum parallel to the nanowire axis, and a change of 0 or ±1 in the total-angular-momentum projection parallel to the nanowire axis. The magnitude of the optical-transition matrix elements is strongly affected by the finite radius of the nanowire.

The two-mass contribution to the three-loop polarized gluonic operator matrix element [formula omitted

We compute the two-mass contributions to the polarized massive operator matrix element Agg,Q(3) at third order in the strong coupling constant αs in Quantum Chromodynamics analytically. These corrections are important ingredients for the matching relations in the variable flavor number scheme and for the calculation of Wilson coefficients in deep–inelastic scattering in the asymptotic regime Q2≫mc2,mb2. The analytic result is expressed in terms of nested harmonic, generalized harmonic, cyclotomic and binomial sums in N-space and by iterated integrals involving square-root valued arguments in z space, as functions of the mass ratio. Numerical results are presented. New two–scale iterative integrals are calculated.

The O(α2) initial state QED corrections to [formula omitted

We calculate the complete O(α2) initial state radiation corrections to e+e− annihilation into a neutral vector boson in a direct analytic computation without any approximation. The corrections are represented in terms of iterated incomplete (elliptic) integrals over alphabets of square-root valued letters. Performing the limit s≫me2, we find discrepancies with the earlier results of Ref. [1] and confirm results obtained in Ref. [2] where the effective method of massive operator matrix elements has been used, which works for all but the power corrections in me2/s. In this way, we also confirm the validity of the factorization of massive partons in the Drell-Yan process to O(α2). We add non–logarithmic terms at O(α2) which have not been considered in previous calculations. The final results in the limit s≫me2 can be given in terms of Nielsen integrals.

Subleading logarithmic QED initial state corrections to [formula omitted] to O(α6L5)

Using the method of massive operator matrix elements, we calculate the subleading QED initial state radiative corrections to the process e+e−→γ⁎/Z⁎ for the first three logarithmic contributions from O(α3L3),O(α3L2),O(α3L) to O(α5L5),O(α5L4),O(α5L3) and compare their effects to the leading contribution O(α6L6) and one more subleading term O(α6L5). The calculation is performed in the limit of large center of mass energies squared s≫me2. These terms supplement the known corrections to O(α2), which were completed recently. Given the high precision at future colliders operating at very large luminosity, these corrections are important for concise theoretical predictions. The present calculation needs the calculation of one more two–loop massive operator matrix element in QED. The radiators are obtained as solutions of the associated Callen–Symanzik equations in the massive case. The radiators can be expressed in terms of harmonic polylogarithms to weight w = 6 of argument z and (1−z) and in Mellin N space by generalized harmonic sums. Numerical results are presented on the position of the Z peak and corrections to the Z width, ΓZ. The corrections calculated result into a final theoretical accuracy for δMZ and δΓZ which is estimated to be of O(30keV) at an anticipated systematic accuracy at the FCC_ee of ∼100keV. This precision cannot be reached, however, by including only the corrections up to O(α3).

Wigner function of a quantum system with polynomial potential

The Moyal equation for the Wigner function was obtained under the assumption that the potential is an analytic function. The polynomial form of the potential is a natural approximation of the analytical potential with any necessary accuracy. The simplest quantum system with a second-order polynomial potential is a quantum harmonic oscillator. In this paper, for a quantum system with a polynomial potential of arbitrary order, explicit expressions are obtained for the matrix elements of the kernel operator in the basis of the eigenfunctions of the harmonic oscillator. Using the explicit representation for the kernel operator matrix elements, we construct the distributions of the Wigner function in the phase space for quantum systems with polynomial potentials. The connection of the modified Vlasov equation with the Moyal equation for the Wigner function is shown. Examples of effective numerical algorithms for finding Wigner functions with high accuracy are given.