Reduced Density Matrix of Permutational Invariant Many-body Systems
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Reduced Density Matrix of Permutational Invariant Many-body Systems Mario Salerno · Vladislav Popkov
Received: 6 October 2009 / Accepted: 2 September 2010 / Published online: 17 September 2010 © Springer Science+Business Media B.V. 2010
Abstract We consider density matrices which are sums of projectors on states spanning irreducible representations of the permutation group of L sites (eigenstates of permutational invariant quantum system with L sites) and construct reduced density matrix ρn for blocks of size n < L by tracing out L − n sites, viewed as environment. Explicit analytic expressions of the elements of ρn are given in the natural basis and the corresponding spectrum of the reduced density matrix is derived. Results apply to other quantum many-body systems with permutational symmetry. Keywords Reduced density matrix · Entanglement · von Neuman entropy · Many-body systems
1 Introduction The study of the entanglement properties of interacting quantum many-body systems [1] is presently receiving a great deal of attention due to its relevance for emergent fields like quantum computation and quantum cryptography [2]. Entanglement properties have been investigated for several spin chains [3–11], for strongly correlated fermions [6, 12, 13] and pairing models [14–16], for itinerant bosons [17], etc. The calculation of the entanglement involves the knowledge of the reduced density matrix (RDM) characterizing open quantum systems, i.e. quantum systems in contact with an environment such as a thermal bath or a larger system of which the original system constitutes a part (subsystem). The spectrum of the RDM, which, as for any density matrix is real, nonnegative with all eigenvalues summing up to one, carries important information about the subsystem. The most representative states of the subsystem, indeed, are the eigenstates of the RDM associated to the largest eigenvalues. This propriety is often used in numerical algorithms for real M. Salerno () · V. Popkov Dipartimento di Fisica “E.R. Caianiello” and Instituto Nazionale di Fisica Nucleare (INFN), Gruppo Collegato di Salerno, and Consorzio Nazionale per le Scienze Fisiche della Materia (CNISM), Universitá di Salerno, via Ponte don Melillo, 84084 Fisciano, SA, Italy e-mail: [email protected]
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M. Salerno, V. Popkov
space renormalization group calculations of quantum many-body systems [18, 19]. Thus, for example, the fact that the RDM eigenvalues λi decay exponentially with i for one dimensional quantum interacting subsystems, implies that the subsystem can be characterized by only few states, a property which is crucial for the success of the density-matrix renormalization group method (DMRG) [18, 19] in one dimension (the fact that this property is violated in higher dimensions is the main reason for the failure of the DMRG in these cases [20]). For a subsystem consisting of n sites the RDM is a matrix of size 2n × 2n so that for large n the calculation of the spectrum becomes a problem of exponential difficulty. While the spectrum of the full RDM for s
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