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arXiv:2301.01419

Reduced Density Matrices and Moduli of Many-Body Eigenstates

Chaoming Song

Published Jan 4, 2023• 0 citations
arXivHTMLPDF

Abstract

Many-body wavefunctions usually lie in high-dimensional Hilbert spaces. However, physically relevant states, i.e, the eigenstates of the Schrödinger equation are rare. For many-body systems involving only pairwise interactions, these eigenstates form a low-dimensional subspace of the entire Hilbert space. The geometry of this subspace, which we call the eigenstate moduli problem is parameterized by a set of 2-particle Hamiltonian. This problem is closely related to the $N$-representability conditions for 2-reduced density matrices, a long-standing challenge for quantum many-body systems. Despite its importance, the eigenstate moduli problem remains largely unexplored in the literature. In this Letter, we propose a comprehensive approach to this problem. We discover an explicit set of algebraic equations that fully determine the eigenstate spaces of $m$-interaction systems as projective varieties, which in turn determine the geometry of the spaces for representable reduced density matrices. We investigate the geometrical structure of these spaces, and validate our results numerically using the exact diagonalization method. Finally, we generalize our approach to the moduli problem of the arbitrary family of Hamiltonians parameterized by a set of real variables.

DOI
10.48550/arxiv.2301.01419

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Many-body wavefunctions usually lie in high-dimensional Hilbert spaces. However, physically relevant states, i.e, the eigenstates of the Schrödinger equation are rare. For many-body systems involving only pairwise interactions, these eigenstates form a low-dimensional subspace of the entire Hilbert space. The geometry of this subspace, which we call the eigenstate moduli problem is parameterized by a set of 2-particle Hamiltonian. This problem is closely related to the NNN-representability conditions for 2-reduced density matrices, a long-standing challenge for quantum many-body systems. Despite its importance, the eigenstate moduli problem remains largely unexplored in the literature. In this Letter, we propose a comprehensive approach to this problem. We discover an explicit set of algebraic equations that fully determine the eigenstate spaces of mmm-interaction systems as projective varieties, which in turn determine the geometry of the spaces for representable reduced density matrices. We investigate the geometrical structure of these spaces, and validate our results numerically using the exact diagonalization method. Finally, we generalize our approach to the moduli problem of the arbitrary family of Hamiltonians parameterized by a set of real variables.
It is virtually impossible to directly solve the Schrödinger equation for a many-electron wave function due to the exponential growth in degrees of freedom with increasing particle number. The two-body reduced density matrix (2-RDM) formalism reduces this coordinate dependence to that of four particles irrespective of the wave function's dimensionality, providing a promising path to solve the many-body problem. Unfortunately, errors arise in this approach because the 2-RDM cannot practically be constrained to guarantee that it corresponds to a valid wave function. Here we approach this so-called NNN-representability problem by expanding the 2-RDM in a complete basis of two-electron wave functions and studying the matrix formed by the expansion coefficients. This quantity, which we call the geminal density matrix (GDM), is found to evolve in time by a unitary transformation that preserves NNN-representability. This evolution law enables us to calculate eigenstates of strongly correlated systems by a fictitious adiabatic evolution in which the electron-electron interaction is slowly switched on. We show how this technique is used to diagonalize atomic Hamiltonians, finding that the problem reduces to the solution of ∼N(N−1)/2\sim N(N-1)/2∼N(N−1)/2 two-electron eigenstates of the Helium atom on a grid of electron-electron interaction scaling factors.
The reduced kkk-particle density matrix of a density matrix on finite-dimensional, fermion Fock space can be defined as the image under the orthogonal projection in the Hilbert-Schmidt geometry onto the space of kkk-body observables. A proper understanding of this projection is therefore intimately related to the representability problem, a long-standing open problem in computational quantum chemistry. Given an orthonormal basis in the finite-dimensional one-particle Hilbert space, we explicitly construct an orthonormal basis of the space of Fock space operators which restricts to an orthonormal basis of the space of kkk-body operators for all kkk.
The description of quantum many-body systems is an exceedingly cumbersome challenge since the dimensions of their Hilbert spaces scale exponentially with the number of constituents. Whereas mean-field approaches are expected to provide accurate descriptions for high dimensional lattices, numerical approaches based on the density matrix renormalization group have very successfully been applied to one-dimensional chains. Yet for two-dimensional lattices there is still a pressing need for efficient descriptions as these are expected to host some very intriguing but still illusive phenomena in quantum many-body physics. Here we derive an exact equation of motion for the reduced density matrices of individual subsystems of quantum many-body systems of any lattice dimension and arbitrary system size. Our projection operator based approach is thus highly efficient and not restricted to unitary dynamics. We show its excellent accuracy for moderate size one-dimensional and small two-dimensional systems where very accurate or exact numerical solutions are available.
In this thesis, we will show how certain classes of quantum many-body Hamiltonians with su21oplussu22oplusldotsoplussu2k\\su{2}_1 \\oplus \\su{2}_2 \\oplus \\ldots \\oplus \\su{2}_ksu21​oplussu22​oplusldotsoplussu2k​ spectrum generating algebras can be approximated by multi-dimensional shifted harmonic oscillator Hamiltonians. The dimensions of the Hilbert spaces of such Hamiltonians usually depend exponentially on kkk. This can make obtaining eigenvalues by diagonalization computationally challenging. The Shifted Harmonic Approximation (SHA) developed here gives good predictions of properties such as ground state energies, excitation energies and the most probable states in the lowest eigenstates. This is achieved by solving only a system of kkk equations and diagonalizing ktimeskk\\times kktimesk matrices. The SHA gives accurate approximations over wide domains of parameters and in many cases even across phase transitions.\n\nThe SHA is first illustrated using the Lipkin-Meshkov-Glick (LMG) model and the Canonical Josephson Hamiltonian (CJH) which have su2\\su{2}su2 spectrum generating algebras. Next, we extend the technique to the non-compact su1,1\\su{1,1}su1,1 algebra, using the five-dimensional quartic oscillator (5DQO) as an example. Finally, the SHA is applied to a kkk-level Bardeen-Cooper-Shrieffer (BCS) pairing Hamiltonian with fixed particle number. The BCS model has a su21oplussu22oplusldotsoplussu2k\\su{2}_1 \\oplus \\su{2}_2 \\oplus \\ldots \\oplus \\su{2}_ksu21​oplussu22​oplusldotsoplussu2k​ spectrum generating algebra.\n\nAn attractive feature of the SHA is that it also provides information to construct basis states which yield very accurate eigenvalues for low-lying states by diagonalizing Hamiltonians in small subspaces of huge Hilbert spaces. For Hamiltonians that involve a smaller number of operators, accurate eigenvalues can be obtained using another technique developed in this thesis: the generalized Rowe-Rosensteel-Kerman-Klein equations-of-motion method (RRKK). The RRKK is illustrated using the LMG and the 5DQO. In RRKK, solving unknowns in a set of 10times1010\\times 1010times10 matrices typically gives estimates of the lowest few eigenvalues to an accuracy of at least eight significant figures. The RRKK involves optimization routines which require initial guesses of the matrix representations of the operators. In many cases, very good initial guesses can be obtained using the SHA. \n\nThe thesis concludes by exploring possible future developments of the SHA.
The reduction of density matrices defined with respect to a finite basis set is considered. A basis set can be introduced into the vector space of Hermitian matrices acting on functions of the coordinates of ppp particles for each value of ppp. An analogy with the construction of spin eigenfunctions is used to obtain a particular basis set called the reducing basis. Reduction of one of these basis elements is either one to one or maps into the origin. Any element in the preimage of a density matrix can thus be resolved into two components, one uniquely determined by the density matrix and the other arbitrary within a certin subset. A restatement of the NNN-representability problem is given, and two sufficient conditions and one necessary condition for NNN representability are given in terms of distances and norms.
Solving for the lowest energy eigenstate of the many-body Schrodinger equation is a cornerstone problem that hinders understanding of a variety of quantum phenomena. The difficulty arises from the exponential nature of the Hilbert space which casts the governing equations as an eigenvalue problem of exponentially large, structured matrices. Variational methods approach this problem by searching for the best approximation within a lower-dimensional variational manifold. In this work we use graph neural networks to define a structured variational manifold and optimize its parameters to find high quality approximations of the lowest energy solutions on a diverse set of Heisenberg Hamiltonians. Using graph networks we learn distributed representations that by construction respect underlying physical symmetries of the problem and generalize to problems of larger size. Our approach achieves state-of-the-art results on a set of quantum many-body benchmark problems and works well on problems whose solutions are not positive-definite. The discussed techniques hold promise of being a useful tool for studying quantum many-body systems and providing insights into optimization and implicit modeling of exponentially-sized objects.