We report results on various techniques which allow to compute the expansion into Legendre (or in... more We report results on various techniques which allow to compute the expansion into Legendre (or in general Gegenbauer) polynomials in an efficient way. We describe in some detail the algebraic/symbolic approach already presented in Ref.1 and expand on an alternative approach based on a theorem of Schoenberg.
The renormalization group equation describing the evolution of the metric of the non linear sigma... more The renormalization group equation describing the evolution of the metric of the non linear sigma models poses some nice mathematical problems involving functional analysis, differential geometry and numerical analysis. We describe the techniques which allow a numerical study of the solutions in the case of a two-dimensional target space (deformation of the $O(3)\; \sigma$--model. Our analysis shows that the so-called sausages define an attracting manifold in the U(1) symmetric case, at one-loop level. The paper describes i) the known analytical solutions, ii) the spectral method which realizes the numerical integrator and allows to estimate the spectrum of zero--modes, iii) the solution of variational equations around the solutions, and finally iv) the algorithms which reconstruct the surface as embedded in $R^3$.
We compute the complex temperature singularities for the two-dimensional O(N) symmetric Heisenber... more We compute the complex temperature singularities for the two-dimensional O(N) symmetric Heisenberg model in the limit N-, oo. The model is exactly soluble but the analysis of singularities requires a quite arduous study of analytic properties of the inverse of an elliptic function with respect to its modulus. We show that the mass gap and other physical quantities are analytic in a narrow strip around the real inverse temperature axis. This fact explains why low-temperature properties are not easily described by high-temperature expansion methods. We then use the known analyticity properties to identify a conformal transformation which improves the convergence of the high-temperature expansion and, as a consequence, the estimate of physical quantities at low temperature. The method is suitable for extension to all N > 2 models, in which a similar analytic structure presumably holds.
We study the spectrum of the Schroedinger operator for a particle constrained on a two-dimensiona... more We study the spectrum of the Schroedinger operator for a particle constrained on a two-dimensional flat torus under the combined action of a transverse magnetic field and a conservative force. A numerical method is presented which allows to compute the spectrum with high accuracy. The method employs a fast Fourier transform to accurately represent the momentum variables and takes into account the twisted boundary conditions required by the presence of the magnetic field. An accuracy of 12 digits is attained even with coarse grids. Landau levels are reproduced in the case of a uniform magnetic field satisfying Dirac's condition. A new fine structure of levels within the single Landau level is formed when the field has a sinusoidal component with period commensurable to the integer magnetic charge. This fact is interpreted in terms of the peculiar symmetry ZN × ZN which holds in the unperturbed case.
We consider the design of a non-local MonteCarlo algorithm for SU (3) lattice systems according t... more We consider the design of a non-local MonteCarlo algorithm for SU (3) lattice systems according to the idea of embedding the degrees of freedom corresponding to the center of the group Z(3). As a crucial ingredient to reach this goal, we present a practical implementation of a cluster algorithm for Z(3) systems with general random pair interaction.
A new class of time-energy uncertainty relations is directly derived from the Schrödinger equatio... more A new class of time-energy uncertainty relations is directly derived from the Schrödinger equations for time-dependent Hamiltonians. Only the initial states and the Hamiltonians, but neither the instantaneous eigenstates nor the full time-dependent wave functions, which would demand a full solution for a time-dependent Hamiltonian, are required for our time-energy relations. Explicit results are then presented for particular subcases of interest for timeindependent Hamiltonians and also for time-varying Hamiltonians employed in adiabatic quantum computation. Some estimates of the lower bounds on computational time are given for general adiabatic quantum algorithms, with Grover's search as an illustration. We particularly emphasize the role of required energy resources, besides the space and time complexity, for the physical process of (quantum) computation, in general.
We present some evidence that in lattice gauge theory the condensate contains a non-perturbative ... more We present some evidence that in lattice gauge theory the condensate contains a non-perturbative contribution proportional to Λ2, the square of the physical scale. This result is based on an analysis of the Wilson loop plaquette expectation from Monte Carlo simulations and its perturbative expansion computed to eight loops. The analysis is not fully conclusive since the calculations are done
equation for four-point correlation function in Liouville field theory and elliptic four-point co... more equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks
We use the algebraic definition of the Dilatation operator provided by Minahan, Zarembo, Beisert,... more We use the algebraic definition of the Dilatation operator provided by Minahan, Zarembo, Beisert, Kristijansen, Staudacher, proper for single trace products of scalar fields, at leading order in the large-N 't Hooft limit to develop a new approach to the evaluation of the spectrum of the Dilatation operator. We discover a vast number of exact sequences of eigenstates.
We present a short note about the perturbative correction to Rydberg energies under a perturbatio... more We present a short note about the perturbative correction to Rydberg energies under a perturbation cosθ/r μ and discuss the role of SO(4) symmetry.
We report results on various techniques which allow to compute the expansion into Legendre (or in... more We report results on various techniques which allow to compute the expansion into Legendre (or in general Gegenbauer) polynomials in an efficient way. We describe in some detail the algebraic/symbolic approach already presented in Ref.1 and expand on an alternative approach based on a theorem of Schoenberg.
The renormalization group equation describing the evolution of the metric of the non linear sigma... more The renormalization group equation describing the evolution of the metric of the non linear sigma models poses some nice mathematical problems involving functional analysis, differential geometry and numerical analysis. We describe the techniques which allow a numerical study of the solutions in the case of a two-dimensional target space (deformation of the $O(3)\; \sigma$--model. Our analysis shows that the so-called sausages define an attracting manifold in the U(1) symmetric case, at one-loop level. The paper describes i) the known analytical solutions, ii) the spectral method which realizes the numerical integrator and allows to estimate the spectrum of zero--modes, iii) the solution of variational equations around the solutions, and finally iv) the algorithms which reconstruct the surface as embedded in $R^3$.
We compute the complex temperature singularities for the two-dimensional O(N) symmetric Heisenber... more We compute the complex temperature singularities for the two-dimensional O(N) symmetric Heisenberg model in the limit N-, oo. The model is exactly soluble but the analysis of singularities requires a quite arduous study of analytic properties of the inverse of an elliptic function with respect to its modulus. We show that the mass gap and other physical quantities are analytic in a narrow strip around the real inverse temperature axis. This fact explains why low-temperature properties are not easily described by high-temperature expansion methods. We then use the known analyticity properties to identify a conformal transformation which improves the convergence of the high-temperature expansion and, as a consequence, the estimate of physical quantities at low temperature. The method is suitable for extension to all N > 2 models, in which a similar analytic structure presumably holds.
We study the spectrum of the Schroedinger operator for a particle constrained on a two-dimensiona... more We study the spectrum of the Schroedinger operator for a particle constrained on a two-dimensional flat torus under the combined action of a transverse magnetic field and a conservative force. A numerical method is presented which allows to compute the spectrum with high accuracy. The method employs a fast Fourier transform to accurately represent the momentum variables and takes into account the twisted boundary conditions required by the presence of the magnetic field. An accuracy of 12 digits is attained even with coarse grids. Landau levels are reproduced in the case of a uniform magnetic field satisfying Dirac's condition. A new fine structure of levels within the single Landau level is formed when the field has a sinusoidal component with period commensurable to the integer magnetic charge. This fact is interpreted in terms of the peculiar symmetry ZN × ZN which holds in the unperturbed case.
We consider the design of a non-local MonteCarlo algorithm for SU (3) lattice systems according t... more We consider the design of a non-local MonteCarlo algorithm for SU (3) lattice systems according to the idea of embedding the degrees of freedom corresponding to the center of the group Z(3). As a crucial ingredient to reach this goal, we present a practical implementation of a cluster algorithm for Z(3) systems with general random pair interaction.
A new class of time-energy uncertainty relations is directly derived from the Schrödinger equatio... more A new class of time-energy uncertainty relations is directly derived from the Schrödinger equations for time-dependent Hamiltonians. Only the initial states and the Hamiltonians, but neither the instantaneous eigenstates nor the full time-dependent wave functions, which would demand a full solution for a time-dependent Hamiltonian, are required for our time-energy relations. Explicit results are then presented for particular subcases of interest for timeindependent Hamiltonians and also for time-varying Hamiltonians employed in adiabatic quantum computation. Some estimates of the lower bounds on computational time are given for general adiabatic quantum algorithms, with Grover's search as an illustration. We particularly emphasize the role of required energy resources, besides the space and time complexity, for the physical process of (quantum) computation, in general.
We present some evidence that in lattice gauge theory the condensate contains a non-perturbative ... more We present some evidence that in lattice gauge theory the condensate contains a non-perturbative contribution proportional to Λ2, the square of the physical scale. This result is based on an analysis of the Wilson loop plaquette expectation from Monte Carlo simulations and its perturbative expansion computed to eight loops. The analysis is not fully conclusive since the calculations are done
equation for four-point correlation function in Liouville field theory and elliptic four-point co... more equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks
We use the algebraic definition of the Dilatation operator provided by Minahan, Zarembo, Beisert,... more We use the algebraic definition of the Dilatation operator provided by Minahan, Zarembo, Beisert, Kristijansen, Staudacher, proper for single trace products of scalar fields, at leading order in the large-N 't Hooft limit to develop a new approach to the evaluation of the spectrum of the Dilatation operator. We discover a vast number of exact sequences of eigenstates.
We present a short note about the perturbative correction to Rydberg energies under a perturbatio... more We present a short note about the perturbative correction to Rydberg energies under a perturbation cosθ/r μ and discuss the role of SO(4) symmetry.
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Papers by Enrico Onofri