Many body density matrix

Recently, the key problems associated with the failure of the old RG method have been identified and a different renormalization procedure based on the eigenvalues of the many-body density matrix of proper subsystems has been developed [64, 65]. This method has come to be known as the density matrix renormalization group (DMRG) method and has found dramatic success in dealing with... 

The many-body density matrix of a part of the system can be easily constructed as follows. Let us begin with given state ji/ >s of 5, which is called the universe or superblock, consisting of the system (which we call a block) A and its environment A. Let us assume that the Fock space of A and A are known, and can be labelled as i >A... 

Construct the reduced many-body density matrix, p, for the block A. If the system does not possess reflection symmetry, construct the density matrix, p, for the right block A as well. 

At the end of eax h iteration, one can calculate the properties of the targeted state [67]. The reduced many-body density matrix computed at each iteration can be used to calculate the static expectation values of any site operator or their products. Care should be taken to use the density matrices appropriate to the iteration. The expectation value of a site property corresponding to the operator Ai can be written as ... 

To summarize, we have presented a review on the renormalization group theory in the reduced many-body density matrix basis (DMRG method), and we have applied it to conjugated organic systems, with both short range and long range Coulomb interaction potentials. We... 

M. Rosina, Direct variational calculation of the two-body density matrix, in The Nuclear Marty-Body Problem Proceedings the Symposium on Present Status and Novel Developments in the Nuclear Many-Body Problem, Rome 1972, (F. Calogero and C. Ciofi degli Atti, eds.), Editrice Compositori, Bologna, 1973. 

Keywords strongly correlated electrons nondynamic correlation density matrix renormalization group post Hartree-Fock methods many-body basis matrix product states complete active space self-consistent field electron correlation... 

If we are interested in accurate properties of the system at a chosen size, then it is possible to improve upon the accuracies obtainable from the infinite DMRG procedure. This involves recognizing that the reduced many-body density matrices at each iteration correspond to a different system size. For example, when we carry out the DMRG procedure to obtain the properties of a system of 2M sites, at an iteration corresponding to 2p sites n < p < M), the reduced density matrix we construct is that of a block of p sites in a system of 2p sites. However, if our interest is in the 2M-site system, we should employ the density matrix of the block of p sites in a 2M-site system. It is possible to construct, iteratively, the p-site reduced density matrix of the 2M-site system. This is achieved by the so called finite-size algorithm. 

In the same manner can be constructed also the higher orders of semiclas-sical expansion of density matrix (3.48) or (3.54), following the cumu-lant expansion (3.118), its fluctuation path and connected Green function components, as given by Eqs. (3.96) and (251), respectively, towards constructing the analytical canonical density, partition function and finally the many-body density to be used in DFT and of its (chemical) applications. 

Tredgold, R. H., Phys. Rev. 105, 1421, "Density matrix and the many-body problem."... 

In Bohmian mechanics, the way the full problem is tackled in order to obtain operational formulas can determine dramatically the final solution due to the context-dependence of this theory. More specifically, developing a Bohmian description within the many-body framework and then focusing on a particle is not equivalent to directly starting from the reduced density matrix or from the one-particle TD-DFT equation. Being well aware of the severe computational problems coming from the first and second approaches, we are still tempted to claim that those are the most natural ways to deal with a many-body problem in a Bohmian context. 

Decius, J. C. 1963. Compliance matrix and molecular vibrations. J. Chem. Phys. 38 241-248. Dreizler R. M. and E. K. U. Gross. 1990. Density Functional Theory An Approach to the Quantum Many-Body Problem. Berlin Springer-Verlag. 

But there is a more basic difficulty in the Hohenberg-Kohn formulation [19-21], which has to do with the fact that the functional iV-representability condition on the energy is not properly incorporated. This condition arises when the many-body problem is presented in terms of the reduced second-order density matrix in that case it takes the form of the JV-representability problem for the reduced 2-matrix [19, 22-24] (a problem that has not yet been solved). When this condition is not met, an energy functional is not in one-to-one correspondence with either the Schrodinger equation or its equivalent variational principle therefore, it can lead to energy values lower than the experimental ones. 

Then, in the Old Ages (1940 or 1951-1967) some ingenious people became aware that, in the case of two-body interactions, it is the two-particle reduced density matrix (2-RDM) that carries in a compact way all the relevant information about the system (energy, correlations, etc.). Early insight by Husimi (1940) and challenges by Charles Coulson were completed by a clear realization and formulation of the A-representability problem by John Coleman in 1951 (for the history, see his book [1] and Chapters 1 and 17 of the present book). Then a series of theorems on A-representability followed, by John Coleman and many... 

We have focused on the lower bound method, but density matrix research has moved forward on a much broader front than that. In particular, work on the contracted Schrodinger equation played an important role in developments. A more complete picture can be found in Coleman and Yukalov s book [23]. It has taken 55 years and work by many scientists to fulfill Coleman s 1951 claim at Chalk River that except for a few details which would be easily overcome in a couple of weeks—the A-body problem has been reduced to a 2.5-body problem ... 

There are useful two- and many-electron analogues of the functions discussed above, but when the Hamiltonian contains only one- and two-body operators it is sufficient to consider the pair functions thus the analogue of p(x x ) is the pair density matrix 7t(xi,X2 x i,x ) while that of which follows on identifying and integrating over spin variables as in (4), is H(ri,r2 r i,r2)- When the electron-electron interaction is purely coulombic, only the diagonal element H(ri,r2) is required and the expression for the total interaction energy becomes... 

We have chosen here to hint at the density matrix concept, typical of many-body theories, in order to stress that E c is still a two-electrons operator. (For the many body derivation of (20), see ). 

F.6), there appears the possibility to consider the latter to be the reduction of a many-body fermionic pure state to an N-representable two-matrix. Since the density matrix above, if adapted appropriately, consequently is essentially N-representable through its relation to Coleman s extreme case [107], one might, via appropriate projections, completely recover the proper information, cf. corresponding partitioning procedures depicted in Appendix A. The structure described here is also of fundamental importance in connection with the phenomena of superconductivity and superfluidity through its intimate connections with Yang s concept of ODLRO [106], see more under Section 3.2. 

It is important to note, that the single-particle density matrix (195) should not be mixed up with the density matrix in the basis of many-body eigenstates. 

The main goal in the development of mixed quantum classical methods has as its focus the treatment of large, complex, many-body quantum systems. While applications to models with many realistic elements have been carried out [10,11], here we test the methods and algorithms on the spin-boson model, which is the standard test case in this field. In particular, we focus on the asymmetric spin-boson model and the calculation of off-diagonal density matrix elements, which present difficulties for some simulation schemes. We show that both of the methods discussed here are able to accurately and efficiently simulate this model. 

Moszynski R, Jeziorski B, Rybak S, Szalewicz K, Williams HL (1994) Many-body theory of exchange effects in intermolecular interactions. Density matrix approach and applications to He-F-, He-HF, H2-HF, and Ar-H2 dimers. J Chem Phys 100 5080-5092... 

Before discussing general issues related to the gravitational field in more detail, we will mention the following aspects of the development identified above. To formulate a consistent many-body approach, which rests on Eq. (18), we will review some prerequisites. We start with a brief analysis of the many-particle fermion density matrix. 

In this review, we begin with a treatment of the functional theory employing as basis the maximum entropy principle for the determination of the density matrix of equilibrium ensembles of any system. This naturally leads to the time-dependent functional theory which will be based on the TD-density matrix which obeys the von Neumann equation of motion. In this way, we present a unified formulation of the functional theory of a condensed matter system for both equilibrium and non-equilibrium situations, which we hope will give the reader a complete picture of the functional approach to many-body interacting systems of interest to condensed matter physics and chemistry. 

Theorem We consider the density matrix written in terms the many-body states which minimizes the free energy associated with the Hamiltonian... 

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