Unphysical States Research Articles

Filling and temperature dependence of the spin susceptibility of the two-dimensional Hubbard model in the superconducting d-wave phase

The two-dimensional Hubbard model in the superconducting d-wave phase has been solved within the Composite Operator Method (COM) in the two-pole approximation [1]. The unknowns of the theory have been computed exploiting the operatorial relations, dictated by the Pauli principle, existing between the composite operators belonging to the basis. Such a procedure has also allowed to correctly fix the Hilbert space of the problem avoiding to average on unphysical states and permitting to obtain a very good qualitative agreement with the experimental results available in the literature as regards the phase diagram in the T–n plane. Given such an encouraging result, in this short manuscript, we have analyzed the filling and temperature dependence of the static and uniform spin susceptibility, as obtained by means of the COM within a one-loop-like approximation, and we have compared them to the experimentally observed ones for single-layer high-Tc cuprate superconductors.

Prediction of Folding Equilibria of Differently Substituted Peptides Using One-Step Perturbation

Computer simulation using long molecular dynamics (MD) can be used to simulate the folding equilibria of peptides and small proteins. However, a systematic investigation of the influence of the side-chain composition and position at the backbone on the folding equilibrium is computationally as well as experimentally too expensive because of the exponentially growing number of possible side-chain compositions and combinations along the peptide chain. Here, we show that application of the one-step perturbation technique may solve this problem, at least computationally; that is, one can predict many folding equilibria of a polypeptide with different side-chain substitutions from just one single MD simulation using an unphysical reference state. The methodology reduces the number of required separate simulations by an order of magnitude.

Artificial damping in the Kadanoff-Baym dynamics of small Hubbard chains

We perform a comparative study of exact and approximate time-evolved densities in small Hubbard chains. The approximate densities are obtained via many-body perturbation theory (Hartree-Fock, 2nd Born, GW and T-matrix approximations) within the framework of the time-dependent Kadanoff-Baym equations. Benchmarking approximate results against exact ones allows us to address two rather fundamental issues in the non equilibrium dynamics of strongly correlated systems. I) A characterisation of the performance of several standard MBAs in the non-equilibrium regime. Having a definite notion of how good a specific MBA can be is highly relevant to its application to cases (typically, infinite systems) where exact solutions are not available. Our results show that the T-matrix approximation is overall superior to the other MBAs, at all electron densities. II) A scrutiny of the whole idea of Many Body Perturbation Theory in the Kadanoff-Baym sense, when applied to finite systems. The surprising outcome of our study is that during the time evolution, the KBE develop an unphysical steady state solution. This is a genuinely novel feature of the time-dependent KBE, i.e. is not inherited from possible limitations/approximations in the calculation of the initial state. Our extensive numerical characterisation gives robust evidence that the problem occurs in general, whenever MBPT is applied to finite systems, and approximate self energies based upon infinite partial summations are used. We also offer some more conceptual and general consideration on the dependence of this behaviour on the number of particles and system size. This is followed by our conclusions and glimpses of future work.

Decoupling of unphysical states in the minimal pure spinor formalism I

This is the first of a series of two papers where decoupling of unphysical states in the minimal pure spinor formalism is investigated. The multi-loop amplitude prescription for the minimal pure spinor superstring formulated in hep-th/0406055 involves the insertion of picture changing operators in the path integral. These operators are BRST closed in a distributional sense and depend on a number of constant tensors. One can trace the origin of these insertions to gauge fixing, so the amplitudes are formally independent of the constant tensors. We show however by explicit tree-level and one-loop computations that the picture changing operators are not BRST closed inside correlators and the amplitudes do depend on these constant tensors. This is due to the fact that the gauge fixing condition implicit in the existing minimal amplitude prescription is singular and this can lead to Lorentz violation and non-decoupling of BRST exact states. As discussed in hep-th/0406055, a manifestly Lorentz invariant prescription can be obtained by integrating over the constant tensors and in the sequel to this paper, it is shown that when one includes these integrations unphysical states do decouple to all orders despite the fact that the PCO's are not BRST closed inside correlators.

Using one-step perturbation to predict the folding equilibrium of differently stereochemically substituted β-peptides

The one-step perturbation technique is used to predict the folding equilibria for 16 peptides with different stereochemical side-chain substitutions through one or two long-time simulations, one of an unphysical reference state and another of one of the 16 peptides for which many folding events can be sampled. The accuracy of the one-step perturbation results was investigated by comparing to results available from long-time MD simulations of particular peptides. Their folding free energies were reproduced within statistical accuracy. The one-step perturbation results show that an axial substitution at either the C(α) or the C(β) position destabilizes the 3(14)-helical conformation of the hepta-β-peptide, which is consistent with data inferred from experimental CD spectra. The methodology reduces the number of required separate simulations by an order of magnitude.

Successes and Failures of Kadanoff-Baym Dynamics in Hubbard Nanoclusters

We study the nonequilibrium dynamics of small, strongly correlated clusters, described by a Hubbard Hamiltonian, by propagating in time the Kadanoff-Baym equations within the Hartree-Fock, second Born, GW, and T-matrix approximations. We compare the results to exact numerical solutions. We find that the time-dependent T matrix is overall superior to the other approximations, and is in good agreement with the exact results in the low-density regime. In the long time limit, the many-body approximations attain an unphysical steady state which we attribute to the implicit inclusion of infinite-order diagrams in a few-body system.

Decoupling of unphysical states in the minimal pure spinor formalism II

This is the first of a series of two papers where decoupling of unphysical states in the minimal pure spinor formalism is investigated. The multi-loop amplitude prescription for the minimal pure spinor superstring formulated in hep-th/0406055 involves the insertion of picture changing operators in the path integral. These operators are BRST closed in a distributional sense and depend on a number of constant tensors. One can trace the origin of these insertions to gauge fixing, so the amplitudes are formally independent of the constant tensors. We show however by explicit tree-level and one-loop computations that the picture changing operators are not BRST closed inside correlators and the amplitudes do depend on these constant tensors. This is due to the fact that the gauge fixing condition implicit in the existing minimal amplitude prescription is singular and this can lead to Lorentz violation and non-decoupling of BRST exact states. As discussed in hep-th/0406055, a manifestly Lorentz invariant prescription can be obtained by integrating over the constant tensors and in the sequel to this paper, it is shown that when one includes these integrations unphysical states do decouple to all orders despite the fact that the PCO's are not BRST closed inside correlators.

XYmodel on the circle: Diagonalization, spectrum, and forerunners of the quantum phase transition

We exactly diagonalize the finite-size XY model with periodic boundary conditions and analytically determine the ground state energy. We show that there are two types of fermions: singles and pairs, whose dispersion relations have a completely arbitrary gauge-dependent sign. It follows that the ground state is determined by a competition between the vacuum states (with a suitable gauge) of two parity sectors. We finally exhibit some points in finite size systems that forerun criticality. They are associated to single Bogoliubov fermions and to the level crossings between physical and unphysical states. In the thermodynamic limit they approach the ground state and build up singularities at logarithmic rates.

Interface propagation and microstructure evolution in phase field models of stress-induced martensitic phase transformations

Analytical solutions for diffuse interface propagation are found for two recently developed Landau potentials that account for the phenomenology of stress-induced martensitic phase transformations. The solutions include the interface profile and velocity as a function of temperature and stress tensor. An instability in the interface propagation near lattice instability conditions is studied numerically. The effect of material inertia is approximately included. Two methods for introducing an athermal interface friction in phase field models are discussed. In the first method an analytic expression defines the location of the diffuse interface, and the rate of change of the order parameters is required to vanish if the driving force is below a threshold. As an alternative and more physical approach, we demonstrate that the introduction of spatially oscillatory stress fields due to crystal defects and the Peierls barrier, or to a jump in chemical energy, reproduces the effect of an athermal threshold. Finite element simulations of microstructure evolution with and without an athermal threshold are performed. In the presence of spatially oscillatory fields the evolution self-arrests in realistic stationary microstructures, thus the system does not converge to an unphysical single-phase final state, and rate-independent temperature- and stress-induced phase transformation hysteresis are exhibited.

Schrödinger propagation of initial discontinuities leads to divergence of moments

We show that the large phase expansion of the Schrödinger propagation of an initially discontinuous wave function leads to the divergence of average energy, momentum, and displacement, rendering them unphysical states. If initially discontinuous wave functions are considered to be approximations to continuous ones, the determinant of the spreading rate of these averages is the maximal gradient of the initial wave function. Therefore a dilemma arises between the inclusion of discontinuous wave functions in quantum mechanics and the requirement of finite moments.

Excited-state spin-contamination in time-dependent density-functional theory for molecules with open-shell ground states

While most applications of the linear response formulation of time-dependent density-functional theory (TDDFT) have been to the calculation of the excited states of molecules with closed-shell ground states, Casida’s formulation of TDDFT opened the way to TDDFT calculations on molecules with open-shell ground states by allowing for different-orbitals-for-different-spin, Although a number of publications have now appeared applying TDDFT to molecules with open-shell ground states and give surprisingly good results for simple excitations, it is relatively easy to show that some excited states of open-shell molecules will have unphysically large amounts of spin contamination. There is thus a clear need for computational tools which can separate physical from unphysical excited spin states in TDDFT. We address this need by using analytic derivative techniques to develop formulae for the 1- and 2-electron reduced density difference matrices, in essential agreement with those obtained by Rowe in the equation-of-motion superoperator approach to Green’s functions in nuclear physics. The corresponding formula for excited-state spin contamination appears to be generally good enough for assigning excited-state spin symmetries, but does lead to a small overestimation of S ^ 2 in the cases considered here. This (apparently small) problem is eliminated when the Tamm–Dancoff approximation is used in TDDFT.

All-temperature magnon theory of ferromagnetism

We present an all-temperature magnon formalism for ferromagnetic solids. To ourknowledge, this is the first time that all-temperature spin statistics have been calculated.The general impression up to now is that the magnon formalism breaks down atthe Curie point as it introduces a series expansion and unphysical states. Ourtreatment is based on an accurate quantum mechanical representation of theHolstein–Primakoff transformation. To achieve this end, we introduce the ‘Kubo operator’.The treatment is valid for all 14 types of Bravais lattices, and not limited to simple cubicunit cells. In the present work, we carry out a zeroth-order treatment involvingall possible spin states, and leaving out all unphysical states. In a subsequentpaper we will show that the perturbed energy values are very different, but themagnetic properties undergo only small modifications from the zeroth-order results.

Krein space quantization of linear gravity in de Sitter spacetime and generalization to QFT

It has been shown that the negative-norm states or negative-frequency solutions of the field equation are indispensable for a fully covariant quantization (Krein space quantization) of the minimally coupled free scalar field in de Sitter spacetime. The new method of quantization has been applied for the free boson and spinor fields in Minkowski spacetime, and the basic structure of quantum field theory is reformulated in Krein space. It is seen that the presence of unphysical negative-frequency states plays the role of an automatic renormalization tool for the theory.

Low-temperature magnetization in nanofilms

The low-temperature magnetization of a film was analyzed by the use of exact Bose representation of spin operators that does not suffer from the presence of unphysical states. The magnetization of thin films has exponentially small temperature correctness, so that Dyson’s proof about exponentially small correction coming from two bosons at ideal lattice point cannot be used in film analyses. The main conclusions of this work are that magnetic lattice of a thin film is more rigid than the macroscopic lattice and that the autoreduction process (the three layer film divides into two layer subfilms) takes place in the film.

Turbulence-chemistry interactions in CFD modelling of diesel engines

The use of detailed mechanisms is indispensable in predicting intermediate species in practical combustion devices. Detailed mechanisms describing ignition, flame propagation and pollutant formation typically involve several hundred species and elementary reactions. Their size makes it prohibitive to use them in three-dimensional combustion simulations of real-life equipment. Conventionally reduced mechanisms often fail to predict minor radicals and pollutant precursors. In such situations, the ILDM (Intrinsic Low Dimensional Manifold) method gives an effective way of coupling the details of complex chemistry with the time variations due to turbulence. It is a method of automatic reduction of a detailed mechanism, which assumes local equilibrium with respect to the fastest time-scales identified by a local eigenvector analysis. The KIVA-III code is used to simulate the flow in a single cylinder of a DI diesel engine. A RNG (Renormalized Group Theory) k-ϵ model is added to the code to model the turbulent flow field. Turbulence–chemistry interactions are taken into account by integrating the chemical source terms over a presumed probability density function (PDF). A standard approach can lead to unphysical states and rather poor heat-release and temperature profiles. An improvised approach to eliminate this problem is outlined in this work. Since no equilibrium or partial-equilibrium approach for mechanism reduction is involved, accurate concentrations of intermediate species such as O radicals and soot-precursors (C2H2 and C3H3) can be obtained from the ILDM. This improves the accuracy of pollutant predictions. The goal of this work is to establish the methods used as industrially acceptable ways of coupling the chemistry with turbulence.

Doped carrier formulation of thet−Jmodel: Projection constraint and the effective Kondo-Heisenberg lattice representation

We show that the recently proposed doped carrier Hamiltonian formulation of the t-J model should be complemented with the constraint that projects out the unphysical states. With this new important ingredient, the previously used and seemingly different spin-fermion representations of the t-J model are shown to be gauge related to each other. This new constraint can be treated in a controlled way close to half-filling suggesting that the doped carrier representation provides an appropriate theoretical framework to address the t-J model in this region. This constraint also suggests that the t-J model can be mapped onto a Kondo-Heisenberg lattice model. Such a mapping highlights important physical similarities between the quasi two-dimensional heavy fermions and the high-T$_c$ superconductors. Finally we discuss the physical implications of our model representation relating in particular the small versus large Fermi surface crossover to the closure of the lattice spin gap.

Reliability of the Heitler-London approach for the exchange coupling between electrons in semiconductor nanostructures

We calculate the exchange coupling $J$ between electrons in a double-well potential in a two-dimensional semiconductor environment within the Heitler-London (HL) approach. Two functional forms are considered for the single-well potential. We show that by choosing an appropriate and relatively simple single-electron variational wave function it is possible, within the HL approach, to significantly improve the estimates for $J$. In all cases the present scheme overcomes the artifacts and limitations at short interdot distances, previously attributed to the HL method, where unphysical triplet ground states have been found, and leads to an overall agreement with analytic interpolated expressions for $J$ obtained for a donor-type model potential.

Hybrid modeling and dynamics of a controlled reverse flow reactor

AbstractA hybrid system approach is adopted to study the dynamic behavior of a controlled reverse flow reactor, where the occurrence of flow inversions is caused by a feedback control strategy. With this approach it is analyzed, theoretically and numerically a typical behavior of hybrid systems: Zeno phenomena. Hybrid system theory is also adopted to define a suitable Poincarè map that is a composition of two switching maps. Based on the Poincaré map an efficient methodology is developed for the continuation of limit cycles, and detection of local bifurcations as the set‐point value, and inlet temperature are varied. This analysis shows that Zeno states can be coexisting with symmetric and asymmetric periodic regimes in a wide range of the parameters. The analysis of the nonlinear behavior and Zeno phenomena gives an insight on the limits of some model assumptions, and how these assumptions should be removed to avoid the unphysical response of Zeno executions. Considering a time delay due to the valves in the reactor, the unphysical Zeno state is removed, while Zeno like oscillations are shown to be still persistent. © 2007 American Institute of Chemical Engineers AIChE J, 2007

Spectral-product methods for electronic structure calculations

Progress is reported in development, implementation, and application of a spectral method for ab initio studies of the electronic structure of matter. In this approach, antisymmetry restrictions are enforced subsequent to construction of the many-electron Hamiltonian matrix in a complete orthonormal spectral-product basis. Transformation to a permutation-symmetry representation obtained from the eigenstates of the aggregate electron antisymmetrizer is seen to enforce the requirements of the Pauli principle ex post facto, and to eliminate the unphysical (non-Pauli) states spanned by the product representation. Results identical with conventional use of prior antisymmetrization of configurational state functions are obtained in applications to many-electron atoms. The development provides certain advantages over conventional methods for polyatomic molecules, and, in particular, facilitates incorporation of fragment information in the form of Hermitian matrix representatives of atomic and diatomic operators which include the non-local effects of overall electron antisymmetry. An exact atomic-pair expression is obtained in this way for polyatomic Hamiltonian matrices which avoids the ambiguities of previously described semi-empirical fragment-based methods for electronic structure calculations. Illustrative applications to the well-known low-lying doublet states of the H3 molecule in a minimal-basis-set demonstrate that the eigensurfaces of the antisymmetrizer can anticipate the structures of the more familiar energy surfaces, including the seams of intersection common in high-symmetry molecular geometries. The calculated H3 energy surfaces are found to be in good agreement with corresponding valence-bond results which include all three-center terms, and are in general accord with accurate values obtained employing conventional high-level computational-chemistry procedures. By avoiding the repeated evaluations of the many-centered one- and two-electron integrals required in construction of polyatomic Hamiltonian matrices in the antisymmetric basis states commonly employed in conventional calculations, and by performing the required atomic and atomic-pair calculations once and for all, the spectral-product approach may provide an alternative potentially efficient ab initio formalism suitable for computational studies of adiabatic potential energy surfaces more generally.

Fictive-impurity approach to dynamical mean-field theory: A strong-coupling investigation

Quantum Monte Carlo and semiclassical methods are used to solve two- and four-site cluster dynamical mean-field approximations to the square-lattice Hubbard model at half filling and strong coupling. The dyanmical cluster approximation, cluster dynamical mean-field theory, and fictive-impurity approaches are compared. The energy, spin correlation function, phase boundary, and electron spectral function are computed and compared to available exact results. The comparision permits a quantitative assessment of the ability of the different methods to capture the effects of intersite spin correlations. Two real-space methods and one momentum-space representation are investigated. One of the two real-space methods is found to be significantly worse: in it, convergence to the correct results is found to be slow and, for the spectral function, nonuniform in frequency, with unphysical midgap states appearing. Analytical arguments are presented showing that the discrepancy arises because the method does not respect the pole structure of the self-energy of the insulator. Of the other two methods, the momentum-space representation is found to provide the better approximation to the intersite terms in the energy but neither approximation is particularly acccurate and the convergence of the momentum-space method is not uniform. A few remarks on numerical methods are made.