Bose operators

In this paper we present a set of ID and 2D spin-1/2 models with competing F and AF interactions for which the singlet ground-state wave function can be found exactly. This function has a special form expressed in terms of auxiliary Bose operators. This form of the wave function is similar to the MP one but with infinite matrices. For special values of model parameters it can be reduced to the standard MP form. 

The form of wave function (4) resembles the MP form, but with an infinite matrix which is represented by Bose operators. Therefore, we have to pick out the (0b. .. 0b) element of the matrix product instead of the usual trace in the MP formalism [9, 10], because the trace is undefined in this case. The function 4/ contains components with all possible values of spin S (0 < S < N/2) and, in fact, a fraction of the singlet is exponentially small at large N. This component is filtered out by the operator Pq. 

Generally, the wave functions (4) and (26) resemble the MP form but with infinite matrices represented by the Bose operators. However, in accordance to Maleev s representation in the special cases the infinite matrices formed by the Bose operators b+ and b can be broken off to the size n = 2j + 1, and wave function (26) is reduced to the usual MP form... 

One of these models is the spin- ladder with competing interactions of the ferro- and antiferromagnetic types at the F-AF transition line. The exact singlet ground-state wave function on this line is found in the special form expressed in terms of auxiliary Bose-operators. The spin correlators in the singlet state show double-spiral ordering with the period of spirals equal to the system size. 

The substitution of A in the expression (1.5) leads to the matter-radiation hamiltonian H, with a quadratic form of Bose operators, which is diagonalized in a straightforward manner using Tyablikov s method.21 We shall proceed to the same diagonalization with a previous simplification of the hamiltonian H using the electric dipolar gauge. 

Here a is a dimensionless constant, 5p(R) is the density fluctuation of the medium at the position R (the center of symmetry of the benzoic acid dimer), 0)D is the Debye frequency, and N is the number of acoustic modes, cot = 7 sound k, (bk) is the Bose operator of creation (annihilation of a acoustic phonon with the wave vector k). In the localized representation we have... 

One can introduce the Bose operators of creation (annihilation) of one oscillation, b+(b) ... 

Here Efn(0) refers to the /th excited state of a free molecule in the crystal a n + (aQ is the Bose operator of creation (annihilation) of an intramolecular vibrational excitation in the nth molecule M2(k) refers to the energy of an optical phonon with the wave vector k connected with proton oscillations in the O H O bridge (bk) is the Bose operator of phonon creation (annihilation) and is the coupling energy between the molecular excitation and phonons. 

Here the interaction H(S>k is the energy of an incident photon with the wave vector k, and vk (i.k) the Bose operator of creation (annihilation) of the photon the interaction matrix is... 

A more accurate transition from operators Pn/> Bose operators Bnf, P]xf ls given... 

In consequence, the operators B (k) appear as Bose operators of creation of states with quantum numbers / and wavevectors k. The operators Bf(k) are annihilation operators of those states. 

For the diaganolization of quadratic forms of Bose operators, see Appendix A. 

First let us remark that the replacement of Pauli operators by Bose operators applied in this chapter is only approximate since the occupation numbers of paulions can be either 0 or 1, whereas the occupation numbers for bosons can take all nonnegative integer values 0,1,2,.., .28 Therefore the replacement of the operators Ps and Pj by Bose operators can provoke uncontrolled errors in all... 

We can, however, improve the transition from Pauli operators to Bose operators Bs and B] requiring that for an arbitrary number of bosons the number of paulions will equal 0 or 1 (90). To this aim we write the Pauli operators in the form... 

So we obtained the exact transition from Pauli to Bose operators in the form... 

By application of the exact representation (3.192) the terms in the sum on the r.h.s. with v> 1 can be considered as small operators because the smallness increases with increasing v. Indeed, for Bose operators we have... 

Substituting the expansion (3.197) into (3.19) and (3.29) we obtained the desired expansion of the Hamiltonian in powers with respect to Bose operators, when not only the dynamic, but also the kinetic interaction is taken into account. The new anharmonicity terms do not contain kinematic corrections. The role of this anharmonicity in the theory of third order nonlinear optical effects has been discussed in the article by Ovander (92). 

The operator H, given by eqn (4.14), is quadratic with respect to the Bose operators a j,., SM(k), and >M(k)t. Thus the computation of exciton spectra when the retardation is taken into account is equivalent to the diagonalization of the expression (4.14). In this expression only those amplitudes are connected, for which the wavevectors are equal to k or —k, respectively. Therefore instead of (4.14) it is sufficient to diagonalize the quadratic form... 

The diagonalization of the above quadratic form can be carried out by the transition to new Bose operators p and pp (see Appendix A) ... 

To take the interaction between phonons and photons into consideration, it is necessary to add to the Hamiltonian (6.32), the Hamiltonian Ho(a) of the free field of transverse photons and the Hamiltonian Hint for the interaction of the field of transverse photons with phonons. The linear transformation from the operators a and C to the polariton creation and annihilation operators, i.e. to the operators t(k) and p(k), diagonalizes the quadratic part of the total Hamiltonian. The two-particle states of the crystal, corresponding to the excitation of two B phonons, usually have a small oscillator strength and the retardation for such states can be neglected. In view of the afore-said, the quadratic part of the total Hamiltonian with respect to the Bose operators can be written in the form of the sum H0(B) + where... 

Nonlinear optical effects in crystals can be investigated also microscopically without using the phenomenological Maxwell equations. In the framework of this approach one has to keep, in the Hamiltonian of the crystal (formed, for example, by multilevel molecules), not only quadratic but also terms of third, fourth, etc. order with respect to the Bose amplitudes of excitons and photons. The part of the Hamiltonian which is quadratic with respect to the Bose amplitudes (see Ch. 4), can be diagonalized by making use of new Bose operators s(k) and j(k) (see eqn 4.16) so that... 

To find the eigenstates of the full Hamiltonian, which is the sum of the Hamiltonians (10.32) and (10.33), we introduce new Bose operators and t for the mixed exciton-photon states as follows ... 

When speaking of kinematic interaction, it should be noted that the problem of its separation in connection with the transition from Pauli operators to Bose operators is far from new. This problem arises, in particular, for the Heisenberg Hamiltonian, which corresponds, for example, to an isotropic ferromagnet with spin a = 1/2 when spin waves whose creation and annihilation operators obey Bose commutation relations are introduced. This problem was dealt with by many people, including Dyson (6), who obtained the low-temperature expansion for the magnetization. However, even before Dyson s paper, Van Kranendonk (7) proposed to take into account of the kinetic interaction by starting from a picture where one spin wave produces an obstacle for the passage of another spin wave, since two flipped spins cannot be located at the same site (for Frenkel excitons this means that two excitons cannot be localized simultaneously on one and the same molecule). 

In mathematical language, such an approach means adding to the initial Hamiltonian, in which the Pauli operators are replaced by Bose operators, a term that corresponds to the limiting strong repulsion of two bosons in one site. 

The application of this approach to spin waves was called by Dyson naive and criticized as incorrect and leading to results different from those obtained by him (see (6), the end of 3). We shall show in what follows, however, on the basis of an exact representation of Pauli operators in terms of Bose operators, that the picture described above does take place for Frenkel excitons. This takes place only because the excitation energy A for Frenkel excitons is large compared with the width of the exciton band. As for the spin waves, where the inequality indicated above is not satisfied, the cross-section for the scattering of long-wavelength spin waves by each other can indeed, in agreement with Dyson, differ substantially from a value that follows from the hard sphere approximation (7). 

Substituting (3.192) and (3.193) in (15.5) and going over from Pauli operators to Bose operators we obtain, besides the Hamiltonian in the zeroth approximation (15.7), two types of contributions to the operator of the kinematic exciton-exciton interaction. The terms of the first type are those resulting from the fact, as seen from (3.193), that the operator P PS Ns. These terms are proportional to the excitation energy A and have the form... 

In connection with the aforesaid we take into account the next terms of the kinematic exciton-exciton interaction and consider the part not included in H (see eqn 15.8) which, as H does, results from (15.5) by transition to Bose operators and which is determined by matrix elements Mls s,. ... 

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