The eigenvectors of the particle number operator in second quantization are characterized by the block sparsity of their matrix product state representations. This is shown to generalize to other classes of operators. Imposing block sparsity yields a scheme for conserving the particle number that is commonly used in applications in physics. Operations on such block structures, their rank truncation, and implications for numerical algorithms are discussed. Explicit and rank-reduced matrix product operator representations of one- and two-particle operators are constructed that operate only on the non-zero blocks of matrix product states. Particle number conservation and block structures in matrix product states | Calcolo | Springer Nature Link Skip to main content Particle number conservation and block structures in matrix product states Open access Published: 25 May 2022 Volume 59 , article number  24 ( 2022 ) Cite this article You have full access to this open access article Download PDF Save article View saved research Calcolo Aims and scope Submit manuscript Particle number conservation and block structures in matrix product states Download PDF Abstract The eigenvectors of the particle number operator in second quantization are characterized by the block sparsity of their matrix product state representations. However, for certain types of problems, for instance strongly correlated systems with several competing states of lowest energy, these classical methods typically fail to yield good approximations. Thus more flexible data-sparse parametrizations of the linear combinations of Slater determinants that can be formed from a given finite set of orbitals are of interest. An elegant way of representing such linear combinations of antisymmetric functions is the formalism of second quantization , where wavefunctions are represented in terms of the occupation of each orbital by a particle. Whereas the implementation of wavefunction antisymmetry in such tensor formats is problematic in the real-space representation of wavefunctions, this does not present a problem in the second-quantized representation: the antisymmetry properties are encoded in the representation of operators, and the corresponding occupation numbers describing the wavefunctions can be directly approximated in low-rank tensor formats. However, in contrast to real-space approximations of wavefunctions [ 15 ], where the number of electrons is tied to the spatial dimensionality of the problem, this particle number is not fixed in the second-quantized formulation and thus needs to be prescribed explicitly. For \(X^{\{0\}}_k\) , this number does not change. For the occupied state, in \(X^{\{1\}}_k\) the positions of the blocks correspond to increasing the number of particles by one. Example 3.5 In quantum chemistry, not only the particle number is conserved, but also the numbers of spin-up and spin-down particles. So the MPS is an eigenvector of two associated Laplace-like operators \(\varvec{P}_\mathrm {up}\) and \(\varvec{P}_\mathrm {down}\) . For both cases we have \(n_k = 2\) and \(\lambda _{k,1} = 1\) if k even/odd for the up/down operator and \(\lambda _{k,1} = 0\) otherwise. We denote the blocks representing an unoccupied k -th orbital by and those representing an occupied orbital by For k such that \(N< k < K-N+1\) , which we refer to as the generic case , we have \({\mathcal {K}}_{k-1} = {\mathcal {K}}_k = \{0,\ldots ,N\}\) ; otherwise, the number of particles to the right and to the left of orbital k , and hence the elements of \({\mathcal {K}}_{k-1}\) and \({\mathcal {K}}_k\) , are restricted according to ( 3.9 ). 5 Matrix product operators It is well known that the Hamiltonian ( 1.1 ) commutes with the particle number operator \(\varvec{P}\) [ 18 , §1.3.2]: Lemma 5.1 We have that the Hamiltonian and the particle number operator commute, that is, \(\varvec{H}\varvec{P} = \varvec{P} \varvec{H}\) . Furthermore, all eigenvectors of \(\varvec{H}\) are eigenvectors of \(\varvec{P}\) . Thus \(\varvec{H}\) preserves the particle number of a state as well as its block structure. This means that the application of the Hamiltonian to a block-sparse MPS can be expressed in a matrix-free way, leading to an elegant and efficient algorithmic treatment. 5.1 Compact forms of operators We now turn to the ranks of Hamiltonians as in ( 1.1 ) in second quantization in MPO format. As shown in this section, compared to the number of rank-one terms in the representation ( 1.1 ), one can obtain substantially reduced ranks in MPO representations. The basic mechanism behind this rank reduction is described in [ 6 ] for projected Hamiltonians in the context of DMRG solvers and, in an MPO form for full Hamiltonians similar to the one given here, in [ 7 , 23 ]. In other words, if \(\varvec{x} \in {\mathcal {F}}^K_N\) and if \(\varvec{B}\) is any particle number-preserving operator, then \(\varvec{y} := \varvec{B} \varvec{x} \in {\mathcal {F}}^K_N\) has a representation with block structure according to Corollary  3.4 . However, if \(\mathsf {B}\) is an MPO representation of \(\varvec{B}\) as derived in Sect. This inconsistency can only be resolved by noting that the added particle will be removed further down in the j -th position of the tensor. Additionally, we have 6.1.2 Two-site DMRG The classical (two-site) DMRG [ 19 , 40 ] optimizes two neighboring components at once. This allows for a certain rank-adaptivity in between these components. While this gives the algorithm more flexibility, it also means that the subiterates can leave the fixed-rank manifold and even the tangent space. Nevertheless, we can show that the particle number will be preserved. Copy shareable link to clipboard Provided by the Springer Nature SharedIt content-sharing initiative Keywords Second quantization Particle number conservation Matrix product states Matrix product operators Mathematics Subject Classification 15A69 65F15 65Y20 65Z05 Advertisement

The many-body dispersion (MBD) framework is a successful approach for modeling the long-range electronic correlation energy and optical response of systems with thousands of atoms. Inspired by field theory, here we develop a second-quantized MBD formalism (SQ-MBD) that recasts a system of atomic quantum Drude oscillators in a Fock-space representation. SQ-MBD provides: (i) tools for projecting observables (interaction energy, transition multipoles, polarizability tensors) on coarse-grained representations of the atomistic system ranging from single atoms to large structural motifs, (ii) a quantum-information framework to analyze correlations and (non)separability among fragments in a given molecular complex, and (iii) a path toward the applicability of the MBD framework to molecular complexes with even larger number of atoms. The SQ-MBD approach offers conceptual insights into quantum fluctuations in molecular systems and enables direct coupling of collective plasmon-like MBD degrees of freedom with arbitrary environments, providing a tractable computational framework to treat dispersion interactions and polarization response in intricate systems. Inspired by field theory, here we develop a second-quantized MBD formalism (SQ-MBD) that recasts a system of atomic quantum Drude oscillators in a Fock-space representation. SQ-MBD provides: ( i ) tools for projecting observables (interaction energy, transition multipoles, polarizability tensors) on coarse-grained representations of the atomistic system ranging from single atoms to large structural motifs, ( ii ) a quantum-information framework to analyze correlations and (non)separability among fragments in a given molecular complex, and ( iii ) a path toward the applicability of the MBD framework to molecular complexes with even larger number of atoms. The SQ-MBD approach offers conceptual insights into quantum fluctuations in molecular systems and enables direct coupling of collective plasmon-like MBD degrees of freedom with arbitrary environments, providing a tractable computational framework to treat dispersion interactions and polarization response in intricate systems. The many-body dispersion (MBD) framework models long-range electronic correlation and optical response of molecular systems. Here, the authors present a second-quantized MBD method that opens an efficient path to treating collective quantum fluctuations in molecular complexes with large number of atoms. With this goal in mind, we propose here a second quantization formulation of the MBD model (SQ-MBD) that considerably simplifies the calculation of the fragment contributions to observables stemming from collective MBD modes, enhances physical intuition on how MBD effects operate to connect different length scales in atomistic systems and establishes a strong connection between the MBD method and quantum information theory. Such a modification consists essentially of a reparametrization of the atomic QDOs and the addition of a damping factor to the dipole–dipole interaction matrix. These adjustments serve the dual purpose of avoiding divergences from short-range interactions and correctly reproducing the screening effects induced by the local atomic environment. Fig. 1 Theory and practice of the second quantization formulation of the many-body dispersion (SQ-MBD) method. Panel a illustrates a commutative diagram outlining the connection between the original many-body dispersion (MBD) framework and its second-quantized formalism (SQ-MBD). M and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{{{{{{{{\boldsymbol{M}}}}}}}}}$$\end{document} M ~ matrices denote transformation matrices between first- and second- quantization representations for the atomic QDOs and the collective MBD modes, respectively. Panel b shows the mean excitation numbers of atomic QDOs in the MBD ground state for the supramolecular complex of C 70 fullerene surrounded by a cycloparaphenyl ring composed of 8 units ([8]-CPPA). Panel c shows the normalized covariance matrix of excitation numbers for interacting atomic quantum Drude oscillators (QDOs), decomposed into single Cartesian components. The index 3( A −1) + i is assigned to the QDO associated with the displacement of the Drude particle on the A th atom along the i th Cartesian direction. Source data are provided as Source Data files. Second quantization formulation of the MBD method (SQ-MBD) Although the orthogonal matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{{{{{{\mathcal{O}}}}}}}}$$\end{document} O and the set of eigenenergies \documentclass[12pt]{minimal} Finally, a more advanced analysis can be developed in the same SQ-MBD framework, examining multifragment correlations among QDOs and generalizing mutual information concepts to multipartite-entangled systems 78 , 79 . Discussion In summary, we have presented a formulation of the MBD model in the second quantization picture (SQ-MBD), leading to computational and conceptual insights into coupled QDOs in intricate molecular systems. The presented method allowed us to investigate the ground state of the MBD Hamiltonian in terms of the superposition of QDO excited states. Owing to the Fock space representation in the SQ-MBD framework, it becomes possible to simplify the calculation of expectation values of observables in and between MBD ground and excited states. In fact, the SQ-MBD formalism makes clearer the connection not only between the non-interacting and the interacting ground state for the system of QDOs but also between the atomic QDO excitations and the MBD collective plasmon-like quasiparticle excitations.