Hamiltonian In Fock Space Research Articles

Understanding Partial $\mathcal {P}\mathcal {T}$ Symmetry as Weighted Composition Conjugation in Reproducing Kernel Hilbert Space: An application to Non-hermitian Bose-Hubbard Type Hamiltonian in Fock space

A new kind of symmetry behaviour introduced as partial $\mathcal {P}\mathcal {T}$ -symmetry(henceforth $\partial _{\mathcal {P}\mathcal {T}}$ ) is investigated in a typical Fock space setting. The said Fock space is understood as a Reproducing Kernel Hilbert Space (RKHS). The same kind of symmetry is analysed for a non-hermitian Bose-Hubbard type Hamiltonian (involving two boson operators) along with its eigenstates. The phenomenon of symmetry breaking has also been considered.

Approximate Hamiltonian for baryons in heavy-flavor QCD

Aiming at relativistic description of gluons in hadrons, the renormalization group procedure for effective particles (RGPEP) is applied to baryons in QCD of heavy quarks. The baryon eigenvalue problem is posed using the Fock-space Hamiltonian operator obtained by solving the RGPEP equations up to second order in powers of the coupling constant. The eigenstate components that contain three quarks and two or more gluons are heuristically removed at the price of inserting a gluon-mass term in the component with one gluon. The resulting problem is reduced to the equivalent one for the component of three quarks and no gluons. Each of the three quark–quark interaction terms thus obtained consists of a spin-dependent Coulomb term and a spin-independent harmonic oscillator term. Quark masses are chosen to fit the lightest spin-one quarkonia masses most accurately. The resulting estimates for bbb and ccc states match estimates obtained in lattice QCD and in quark models. Masses of ccb and bbc states are also estimated. The corresponding wave functions are invariant with respect to boosts. In the ccb states, charm quarks tend to form diquarks. The accuracy of our approximate Hamiltonian can be estimated through comparison by including components with two gluons within the same method.

Fluctuations around Hartree states in the mean-field regime

We consider the dynamics of a large system of $N$ interacting bosons in the mean-field regime where the interaction is of order $1/N$. We prove that the fluctuations around the nonlinear Hartree state are generated by an effective quadratic Hamiltonian in Fock space, which is derived from Bogoliubov's approximation. We use a direct method in the $N$-particle space, which is different from the one based on coherent states in Fock space.

Perturbative order analysis of the similarity transformed Hamiltonian in Fock-space coupled cluster theory: difference energy and electric response properties

A perturbative analysis of the ground-state similarity transformed Hamiltonian and its effect on the various Fock-space coupled cluster (FSCC) sectors is presented through calculation of ionisation potential, electron affinity, excitation energies and response properties. Various truncation schemes of the effective Hamiltonian are presented with explicit form of the defining equations. Based on such a truncation, the approximate methods are labelled as FSCC(n), where n represents the correlation energy of the ionised, electron attached or excited states corrected at least up to nth order within coupled cluster singles and doubles scheme (CCSD). A lower scaling CC2 type of approach (abbreviated as FS-CC2) is compared against the group of FSCC(n) methods for energies. Electric response properties have been compared and contrasted for the two lower scaling methods: FSCC(2) and FS-CC2. The various truncated methods are tested for a number of small molecules. The results obtained from a range of truncated methods are compared against full FSCCSD calculations.

Site-occupation constraints in mean-field approaches to quantum spin systems at finite temperature

We study the effect of site occupation on the description of quantum spin systems at finite temperature and mean-field level. We impose each lattice site to be occupied by a single electron. This is realized by means of a specific prescription. The outcome of the prescription is compared to the result obtained by means of a projection procedure that fixes the site occupation to one particle per site, on average. The comparison is performed for different representations of the Hamiltonian in Fock space leading to different types of mean-field solutions. The behavior of order parameters is analyzed for each choice of the mean-field and constraint that fixes the occupation rate at each site. Sizable quantitative differences between the outcomes obtained with the two different constraints are observed.

Some comments on the coupled cluster with generalized singles and doubles (CCGSD) ansatz

In this Letter, we present some comments on the nature of the CCGSD ansatz Ψ = e T Φ for an n-electron wave function, with T a linear combination of general one- and two-particle operators. Nooijen had conjectured that such a parameterization is possible for the exact eigenstate Ψ. We point out that the essential reason for the invalidity of this conjecture lies in the fact that the basis operators into which the Fock space Hamiltonian can be expanded, are not closed under commutation, i.e. do not span a Lie algebra. We give two proofs, based, respectively, on the variation principle and the method of moments, to show this. The variational proof traces the same ground as in an earlier one by Nakatsuji, which did not get the due recognition. We, however, formulate it in such a way that the key role of the algebra of the operators of T becomes transparent. We also discuss a related proof of Mazziotti. Our second proof sheds further light on why the ansatz, while not exact, might still provide an accurate description of a many-electron state, as has been found in some recent computational studies. We also discuss two recent attempts to disprove the exactness of the ansatz, based on a dimensionality argument.

Large-momentum convergence of Hamiltonian bound-state dynamics of effective fermions in quantum field theory

Contributions to the bound-state dynamics of fermions in local quantum field theory from the region of large relative momenta of the constituent particles, are studied and compared in two different approaches. The first approach is conventionally developed in terms of bare fermions, a Tamm-Dancoff truncation on the particle number, and a momentum-space cutoff that requires counterterms in the Fock-space Hamiltonian. The second approach to the same theory deals with bound states of effective fermions, the latter being derived from a suitable renormalization group procedure. An example of two-fermion bound states in Yukawa theory, quantized in the light-front form of dynamics, is discussed in detail. The large-momentum region leads to a buildup of overlapping divergences in the bare Tamm-Dancoff approach, while the effective two-fermion dynamics is little influenced by the large-momentum region. This is illustrated by numerical estimates of the large-momentum contributions for coupling constants on the order of between 0.01 and 1, which is relevant for quarks.

Squeezing operations in Fock space and beyond

Combining the signal and the idler modes we show that each nondegenerate squeezing quadratic Hamiltonian may be transformed into the form of a degenerate squeezing Hamiltonian, if one uses the smeared field formalism. For the case of infinitely many photon modes we discuss the existence of squeezing quadratic Hamiltonians in Fock space. This gives a limitation on the squeezing parameters, which guarantees that all squeezed vacua are representable as vectors in Fock space. If this condition is not satisfied (the case of strong squeezing) then all squeezed vacua are outside the Fock space and mutually inequivalent.

Derivation and pilot application of a scheme for intermediate Hamiltonians in Fock space

A Fock space formulation of the intermediate Hamiltonian approach is derived by introducing shift operators in the equations determining effective Hamiltonians in Fock space. A non-hermitian intermediate Hamiltonian is constructed from the Fock space Bloch equation. An alternative derivation, based on a similarity transformation expression, is presented providing access to hermitian intermediate Hamiltonians. In a pilot application, the potential curves of the two lowest1∑ g + states of the H2 molecule are calculated demonstrating the applicability of the scheme.

The relativistic many body problem in molecular theory

The essential features of current methods to take care of correlation effects in non-relativistic calculations are reviewed, with special attention to configuration interaction (CI) and coupled cluster (CC) methods. It is stressed that most of these methods start from a Fock space Hamiltonian. Then relativistic many-electron Hamiltonians are discussed, especially concerning their accuracy, problems related to the definition of electronic and positronic states and to variational collapse. Special attention is given to `no-pair' Hamiltonians, as well as to various choices of the electron interaction. After some comments on the adaptation of standard methods for electron correlation to the relativistic case, the state of the art of molecular relativistic many-electron theory is described. In an appendix a semiclassical derivation of a relativistic many-electron Hamiltonian is outlined.

Direct calculation of energy differences, such as ionization potentials

Indirect methods for the calculation of energy differences imply that two separate calculations of the involved states are performed first and then the difference is taken. In a ‘direct’ method the transition energy is calculated without an attempt to calculate either of the states between which transition takes place. The theory of direct calculations of energy differences is reviewed with emphasis on ionization potentials and special attention to the method of a common unitary transformation of two model states. Results for the ionization potentials of a few small molecules are presented. The elements of a more general theory are outlined in which the Liouvillean superoperator (and its resolvent) as well as the Fock space formulation of quantum chemistry (especially the concept of effective Hamiltonians in Fock space) play a key role.

Quantum chemistry in Fock space. IV. The treatment of permutational symmetry. Spin-free diagrams with symmetrized vertices

The theory of effective Hamiltonians in Fock space is extended by the use of spin-free basis operators as well as vertices (matrix elements) that are adapted to the permutation group. This reduces the number of the necessary operators considerably, especially for operators of high particle rank. An extension of the generalized Wick theorem of spin-free quantum chemistry to permutation symmetry adapted operators is presented and illustrated graphically. The second order diagrams for the energy and the fully contracted (vacuum) diagrams to third and fourth order are given explicitly. The diagram symmetry group GD that consists of line exchanges at the vertices and permutations of external lines is studied in detail and a recipe for the construction of permutation symmetry adapted diagrams with spin-free symmetrized vertices is derived. The use of permutation symmetry in nonperturbative approaches is discussed and the construction of the matrix elements for the final CI is illustrated. In this formalism there is no need for spin algebra.

Quantum chemistry in Fock space. II. Effective Hamiltonians in Fock space

The concept of an effective Hamiltonian in Fock space is introduced. It is based on the division of the entire one-particle space into subspaces of ‘‘active’’ and ‘‘inactive’’ orbitals. The effective Fock space Hamiltonian has—for active model states—the same eigenvalues as the full Hamiltonian. The theory outlined in this context differs from that of paper I mainly in a different definition of the ‘‘diagonal part’’ of an operator, and in the fact that the ‘‘quasidegenerate case’’ applies throughout. The separation theorem, and as a consequence the connected diagram theorem, is shown to hold, in a more limited sense though, even for those normalizations where it did not in the context of universal wave and energy operators. Unlike in the theory of the ‘‘universal’’ operators of paper I the Fock space and n-particle Hilbert space approaches with analogous normalizations are no longer equivalent. In particular, the Primas normalization with a fully Lie-algebraic structure does not lead to a connected diagram expansion if it is formulated in n-particle Hilbert space, only so in a Fock space formulation. In n-particle Hilbert space with the present definition of the diagonal part of an operator the normalizations b (‘‘canonical’’) and c (‘‘Primas’’) happen to agree. As an alternative to the construction of the wave and energy operator W and L by perturbation theory the nonperturbative approach is presented as a generalization of the coupled-cluster method, in detail both in the intermediate and in the unitary normalization. In the unitary variant only a linear system for σ (the logarithm of the wave operator) has to be solved in order to get L correct through fifth order in perturbation theory with important contributions of higher orders included. A generalization of the Hartree–Fock method to Fock space theory is outlined, which guarantees stationarity of all (active) eigenstates with respect to one-particle transformations. A generalized electron pair theory is also defined. An analysis of the necessary computational steps shows that the nonperturbative approaches do not require significantly more computational effort than perturbation theory to the corresponding order. As a numerical example the H2 molecule in a small basis is discussed.