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arXiv:2312.14275 · Physical review. B./Physical review. B

Two-site reduced density matrix from one- and two-particle Green's functions

Gergő Roósz, Anna Kauch, Frederic Bippus, Daniel Wieser, Karsten Held

Published Aug 8, 2024• 4 citations
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Abstract

Strongly correlated electron systems are challenging to calculate, and entanglement in such systems is not widely analyzed. We present an approach that can be used as a postprocessing step for calculating the two-site reduced density matrix and from it entanglement measures such as the mutual information and entanglement negativity. Input is only the one- and two-particle Green's function, which is the output of numerous many-body methods. As an illustration, we present results for a toy model, the Hubbard model on a $2\ifmmode\times\else\texttimes\fi{}2$ cluster and a six-site ring.

DOI
10.1103/physrevb.110.075115

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Strongly correlated electron systems are challenging to calculate, and entanglement in such systems is not widely analyzed. We present an approach that can be used as a postprocessing step for calculating the two-site reduced density matrix and from it entanglement measures such as the mutual information and entanglement negativity. Input is only the one- and two-particle Green's function, which is the output of numerous many-body methods. As an illustration, we present results for a toy model, the Hubbard model on a 2\ifmmode×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 cluster and a six-site ring.
The study of entanglement in strongly correlated electron systems typically requires knowledge of the reduced density matrix. Here, we apply the parquet dynamical vertex approximation to study the two-site reduced density matrix at varying distance, in the Hubbard model at weak coupling. This allows us to investigate the spatial structure of entanglement in dependence of interaction strength, electron filling, and temperature. We compare results from different entanglement measures, and benchmark against quantum Monte Carlo.
We study the properties of the ground states of the one- and two-dimensional Hubbard models at half-filling and moderate doping using entanglement-based measures, which we calculate numerically using the momentum-space density matrix renormalization group (DMRG). In particular, we investigate quantities such as the single-site entropy and two-site mutual information of single-particle momentum states as well as the behavior of the bipartite subsystem entropy for partitions in momentum space. The distribution of these quantities in momentum space gives insight into the fundamental nature of the ground state, insight that can be used to make contact with weak-coupling-based analytic approaches and to optimize numerical methods, the momentum-space DMRG in particular. We study the site and subsystem entropies as a function of interaction strength UUU and system size. In both the one- and two-dimensional cases, we find that the subsystem entropy scales proportionally to U2{U}^{2}U2 for weak UUU and proportionally to volume. Nevertheless, the optimized momentum-space DMRG can provide variationally accurate results for the two-dimensional Hubbard model at weak coupling for moderate system sizes.
We derive an analytical density functional for the single-site entanglement of the one-dimensional homogeneous Hubbard model by means of an approximation to the linear entropy. We show that this very simple density functional reproduces quantitatively the exact results. We then use this functional as input for a local-density approximation to the single-site entanglement of inhomogeneous systems. We illustrate the power of this approach in a harmonically confined system, which could simulate recent experiments with ultracold atoms in optical lattices as well as in a superlattice and in an impurity system. The impressive quantitative agreement with numerical calculations---which includes reproducing subtle signatures of the particle density stages---shows that our density functional can provide entanglement calculations for actual experiments via density measurements. Next we use our functional to calculate the entanglement in disordered systems. We find that, in contrast with the expectation that disorder destroys the entanglement, there exist regimes for which the entanglement remains almost unaffected by the presence of disordered impurities.
We study entanglement of electron spins in many-body systems based on the Green's function approach. As an application we obtain the two-particle density matrix of a noninteracting electron gas and identify its two-spin density matrix as a Werner state. We calculate entanglement measures, a classical correlation, mutual information, and a pair-distribution function of two electrons at zero and finite temperatures. We find that changes of entanglement measures are proportional to T2{T}^{2}T2 at low temperatures.
We introduce transition operators that in a given basis of the single-site states of a many-body system have a single nonvanishing matrix element and introduce their correlation functions. We show that they fall into groups that decay with the same rate. The mutual information defined in terms of the von Neumann entropy between two sites is given in terms of these so-called generalized correlation functions. We confirm numerically that the long-distance decay of the mutual information follows the square of that of the most slowly decaying generalized correlation function. The main advantage of our procedure is that, in order to identify the most relevant physical processes, there is no need to know a priori the nature of the ordering in the system, i.e., no need to explicitly construct particular physical correlation functions. We explore the behavior of the mutual information and the generalized correlation functions for comformally invariant models and for the SU(n)\text{SU}(n)SU(n) Hubbard model with n=2,3,4n=2,3,4n=2,3,4, and 5, which are, in general, not conformally invariant. In this latter case, we show that for filling f=1/qf=1/qf=1/q and q<nq<nq<n, the ground state consists of highly entangled qqq-site units that are further entangled by single bonds. In addition, we extend the picture of the two-site mutual information and the corresponding generalized correlation functions to the nnn-site case.
One- and two-particle reduced density matrices have been calculated for a ``resonating'' Heitler-London wavefunction, in a system which is a finite model of a two-dimensional square planar lattice structure (4×4 net of one-electron atoms with Born-von Karman boundary conditions). The Heitler-London wavefunction describes the electronic ground state of a system with a single narrow half-filled band (the Mott insulator). Various measures of electron correlation (pair correlation functions, etc.) are defined and calculated. The intent of these calculations is to provide ``empirical'' guidelines for many-body methods for calculation of Green's functions for one and two particles (reduced density matrices are initial values for the respective Green's functions). Results show that the Heitler-London description is incompatible with several assumptions and widely used models for the Mott insulator. In particular it is found that (a) pair correlations in the Heitler-London ground state remain local even when ``resonance'' of an extremely large number of valence-bond structures occurs; the long range antiferromagnetic ordering obtained in several models of the Mott insulator (the ``two-sublattice'' or ``different bands for different spins'' models) does not occur. (b) The density matrix for the Heitler-London state does not exhibit a BCS type ``pair condensation'', as has been conjectured to describe the Mott insulator in some recent work. Our results strongly suggest that if the Heitler-London description is appropriate, the ``Hubbard Hamiltonian'', which includes only intraatomic pair interactions, does not offer an adequate model of the Mott insulator.