Spin correlation function

The matrix elements of the reduced density matrix needed to calculate the entanglement can be written in terms of the spin-spin correlation functions and the average magnetization per spin. The spin-spin correlation functions for the ground state are dehned as [62]... 

Using the definition < A > = Tr(pA), we can express all the matrix elements in the density matrix in terms of different spin-spin correlation functions [62] ... 

Consider a. plane square lattice Ising model with a spin variable s J = 1 associated with the site (i,j) and interactions between nearest-neighbor sites. Kaufman and Onsager3 4 have shown how the spin-spin correlation functions in an infinite lattice,... 

As an application of this theorem let us calculate in leading order the decay of the Ising spin correlation function r(0, N) to its limit r(oo) below Tc. From Eq. (12) and the definitions (42) we find easily... 

As shown in Appendix A, this theorem may be applied to the lsing model to prove that the spin-spin correlation functions F(0, N) and T(N, N) decrease monotonically with separation distance to the limiting long-range order r(oo). 

In order to apply Theorem 4 to prove that the Ising model spin correlation functions E(0, N) and T(Y, N) [see Eq. (11)] decreasemonotonically to the long-range order T(oo) we must study the coefficients p , q , rn, and... 

For infinite systems there is another type of result concerning long-range magnetic order, which is implicated in making a material magnetic in the sense that there be a phase transition at the onset, as well as hysteresis. This Neel-state type of ordering concerns the spin correlation function... 

The paper is organized as follows. In See.2 we consider the frustrated spin chain at F-AF transition point and describe the exact singlet ground-state wave function as well as details of the spin correlation function calculations. We discuss the phase diagram of this model and its magnetic properties in the AF phase. In Sec.3 the special spin ladder will be considered. A two-dimensional frustrated spin model with the exact ground state is considered in Sec.4. Sec. 5 is devoted to the construction of the electronic models with the SB type of wave function. The results of this paper are summarized in Sec.6. 

For the sake of simplicity we show the calculation of the spin correlation function in the symmetric case x = —1/4, when the Hamiltonian (2) takes the form... 

We emphasize that the spin correlation function (SrSj) does not depend on the choice of A for a fixed parameters x, y, because the ground-state wave function of the three-parameter set of Hamiltonians (35) is the same. 

The norm of the wave function (44) and expectation values can be also calculated with the use of the recursion technique developed in [14, 11], Certainly, it gives the same expressions (37,42) for spin correlation functions. 

In the given model there are four spins s = 1/2 at each site, and the enumeration of each spin is determined by the order number of the lattice site to which it belongs and by its own number at this site. The spin correlation function therefore has the form... 

Figure 11 Dependence of the spin correlation function (s3(l)S](2)) on the parameter a.

Our results suggest that the spin correlation functions decay exponentially with a correlation length 1 for an arbitrary parameter a. We also assume that the decay of the correlation function is of the exponential type for the 14 parameter model as well, i.e., for any choice of site spinor I>A/u/p. This assumption is supported in special cases 1) the partition of the system into one-dimensional chains with exactly known exponentially decaying correlation functions 2) the two-dimensional AKLT model, for which the exponential character of the decay of the correlation function has been rigorously proved [32], Further evidence of the stated assumption lies in the numerical results obtained for various values of the parameter in the one-parameter model. 

Hiiigularities occurring for h O r r - Furthermore macroscopic observ-ablcjH like the free energj or the spin correlation functions show scaling in T = r - T(. and /i, most similar to the scaling laws in polymer solutions. 

Dynamical quantities are harder to obtain, since the QMC representations only give access to imaginary-time correlation function. With the exception of measurements of spin gaps, which can be obtained from an exponential decay of the spin-spin correlation function in imaginary time, the measurement of real-time or real-frequency correlation functions requires an ill-posed analytical continuation of noisy Monte Carlo data, for example using the Maximum Entropy Method [46-48]. 

They use the boson representation of fermion operators to evaluate the leading asymptotic terms in spin correlation functions. For example,... 

It has already been known for some time that one-dimensionality may lead to low frequency divergences in spin correlation functions of magnetic insulators /17, 18/. However, the investigation of the spin correlation functions of 1-d conductors has been up to now the subject of much less work /19-21/. It is the purpose of this section to show that a frequency dependence study of T can provide much information in TQ an d HQ. 

The spin correlation functions and their dependence on the distance between sites and the coupling between adjacent sites are of great interest in understanding the range of these correlations. In general, for a closed chain... 

One theory which has been successfully applied to the magnetic properties of R-compound alloys containing non-S-state R-ions is the theory of random magnetic anisotropy (RMA) of Harris et al. (1973). This has been used to calculate the magnetoresistivity of amorphous alloys exhibiting RMA. Bhat-tacharjee and Coqblin (1979) first obtained the static spin-spin correlation function of the RMA model by using a self-consistent two-spin cluster approximation. The quasi-elastic approximation of de Gennes and Friedel is then... 

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