Boson operators models

Let us start by considering the algebraic study of the U(2) model. First, we need to realize this algebra by means of four (n ) operators in terms of two creation-annihilation boson operators f and s (and i and s) ... 

We now show that the algebraic realization of the one-dimensional Morse potential can be adopted as a starting point for recovering this same problem in a conventional wave-mechanics formulation. This will be useful for several reasons (1) The connection between algebraic and conventional coordinate spaces is a rigorous one, which can be depicted explicitly, however, only in very simple cases, such as in the present one-dimensional situation (2) for traditional spectroscopy it can be useful to know that boson operators have a well-defined differential operator counterpart, which will be appreciated particularly in the study of transition operators and related quantities and (3) the one-dimensional Morse potential is not the unique outcome of the dynamical symmetry based on U(2). As already mentioned, the Poschl-Teller potential, being isospectral with the Morse potential in the bound-state portion of the spectrum, can be also described in an algebraic fashion. This is particularly apparent after a detailed study of the differential version of these two anharmonic potential models. Here we limit ourselves to a brief description. A more complete analysis can be found elsewhere [25]. As a... 

As a final example of application of the intensive boson operator technique to the one-dimensional algebraic model, we consider the case of the n m Fermi operator introduced in Section III.E [Eq. (3.128)]. A straightforward use of the aforementioned method leads to the classical potential surface... 

AE(t) represents the energy gap between reactant and product states which depends strongly on the thermal motion of the protein. In the spin-boson model, the two states of electron transfer are described as the spin operator, while thermal vibrations of the protein are accounted for by the boson operators. 

Hereafter we put /ig = 1. Below we express our results in terms of the statistical properties (correlators) of the environment s noise, X(t). Depending on the physical situation at hand, one can choose to model the environment via a bath of harmonic oscillators [6, 3]. In this case the generalized coordinate of the reservoir is defined as X = ]T)Awhere xi are the coordinate operators of the oscillators and Aj are the respective couplings. Eq. 2 is then referred to as the spin-boson Hamiltonian [8]. Another example of a reservoir could be a spin bath [11] 5. However, in our analysis below we do not specify the type of the environment. We will only assume that the reservoir gives rise to markovian evolution on the time scales of interest. More specifically, the evolution is markovian at time scales longer than a certain characteristic time rc, determined by the environment 6. We assume that rc is shorter than the dissipative time scales introduced by the environment, such as the dephasing or relaxation times and the inverse Lamb shift (the scale of the shortest of which we denote as Tdiss, tc [Pg.14]

There is a vast field in chemistry where the spin-boson model can serve practical purposes, namely, proton exchange reactions in condensed media [Borgis and Hynes, 1991 Borgis et al., 1989 Morillo et al., 1989 Morillo and Cukier, 1990 Suarez and Silbey, 1991], The early approaches to this model used a perturbative expansion for weak coupling [Silbey and Harris, 1983], Generally speaking, perturbation theory allows one to consider a TLS coupled to an arbitrary bath via the term ftrz, where / is an operator that acts on the bath variables. The equations of motion in the Heisenberg representation for the operators, daldt = i[H, ], have the form... 

In the interacting boson model-2, low lying collective states of nuclei are described in terms of 12 dynamical bosons [ARI77,0TS78], six proton and six neutron bosons. The six proton and neutron bosons have angular momentum J=0 (s-boson) and J=2 (d-boson). It is convenient to introduce creation (d, s ) (u = 2, 1, 0) and annihilation (du,s) operators. When proton ( tt ) and neutron (v) degrees of freedom are added, the creation and annihilation operators assume an extra label ( tt, v), d s, ... 

It is the group 0(4), containing the collective El operator of this boson model + tt a), that is needed to give enhanced El transitions. In the... 

There are special values of the parameter x for which Eqs.(37) are not valid. For cases of integer or half-integer x = j, which correspond to the special cases of the model (35), in Eq. (27) one can easily recognize Maleev s boson representation of spin S = j operators [28] ... 

The model Hamiltonian [Eq. (3)] defined on a continuum has some exact solutions [35]. These have culminated in what is now known as the bosonization technique, in which the interacting fermion fields can be expressed in terms of boson field operators. This method is reviewed in Refs. 15, 16, and 31. 

An alternative approach is the Slave Boson approximation [21] where the Fermionic operators are defined as the product of Boson and spin operators. There is a constraint with the number of Fermions, n /, and number of Bosons, n n f - - m, = 1. The spin part is treated with a RVB spin model and the charge as a Bose-Einstein condensation problem. These leads to a fractionalization of charges (holons) and spin (spinons), where uncondensed holons exist above the SC domain. The temperature crossover of the spinon pairing and the holon condensation, as a function of doping, is identified as peak in the SC domain. 

Let us now return to the two model Hamiltonians introduced in Section 12.2, and drop from now on the subscripts S and SB from the coupling operators. Using the polaron transformation we can describe both models (12.1), (12.2) and (12.4)-(12.6) in a similar language, where the difference enters in the form ofthe coupling to the boson bath... 

The abstract formalism introduced in this chapter builds the fundament of the theory of extended two-particle Green s functions. Our approach is very general in order to allow for a unified treatment of the different species of extended Green s functions discussed in the main part of this paper. Since the discussed propagators can be applied to a wide variety of physical situations, the emphasis of this chapter lies on the unifying mathematical structure. The formalism is developed simultaneously for (projectile) particles of fermionic and bosonic character. We will define the general extended states which serve to define the primary or model space of the extended Green s functions. We also define the /.j-product under which the previously defined extended states fulfil peculiar orthonormality conditions. Finally we introduce a canonical extension of common Fock-space operators and super-operators to the space of the extended states. 

Hamiltonian of the t-J model (Eq. 3) in terms of the slave-boson and pseudofermion operators has to be replaced by... 

To exemplify the generality of our formulation we will consider M = N/2 bosons (or N fermions) described by a set of n> N/2 localized pair functions or geminals h = hi,h2,...hn obtained from appropriate pairing of one-particle basis spin functions. For simplicity we will briefly overlook the fermionic level. With this somewhat imprecise model we will demonstrate its portrayal of various interesting phenomena via the density operator... 

Here, 1 > and 2 > are the diabatic electronic states (exciton and CT states, respectively), J their electronic coupling parameter, uj the vibrational frequency, 6 and h the boson creation and annihilation operators, and g and g2 the equilibrium position shifts in the excited states 1 > and 2 >, and AE the zeroth order splitting between the two electronic states. The zero energy is set to ( AE - Eg (jS) where ft = 1. and Eg is the ground state electronic energy of a Born-Oppenheimer model. 

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