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Stars & Bars: A Compact Representation for Bosonic Occupation States

Caleb Usadi

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

An efficient representation of occupation states of particles on lattices is a critical tool for the exact diagonalization of bosonic Hamiltonians. The memory demands of the traditional method of representing such systems, as arrays of integers, increases rapidly as system size grows, limiting current studies to approximately 16 particles at unit filling. Representing basis vectors using the stars and bars method of combinatorics allows each basis state to be stored as a single 64 bit integer. This optimally compact representation will enable the analysis of new properties of larger bosonic Hamiltonians, including accessible entanglement, which may be useful in evaluating many-body phases as potential candidate quantum resource states.

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Properties of bosonic atoms in small systems with a periodic quasi one-dimensional circular toroidal lattice potential subjected to rotation are examined by performing exact diagonalization in a truncated many body space. The expansion of the many-body Hamiltonian is considered in terms of the first band Bloch functions, and no assumption regarding restriction to nearest-neighbor hopping (tight-binding approximation) is involved. A finite size version of the zero temperature phase diagrams of Fisher et al. \cite{Fisher} is obtained and the results, in remarkable quantitative correspondence with the results available for larger systems, discussed. Ground state properties relating to superfluidity are examined in the context of two-fluid phenomenology. The basic tool, consisting of the intrinsic inertia associated with small rotation angular velocities in the lab frame, is used to obtain ground state `superfluid fractions' numerically. They are analytically associated with one-body, uniform solenoidal currents in the case of the adopted geometry. These currents are in general incoherent superpositions of contributions from each eigenstate of the associated reduced one-body densities, with the corresponding occupation numbers as weights. Full coherence occurs therefore only when only one eigenstate is occupied by all bosons. The obtained numerical values for the superfluid fractions remain small throughout the parameter region corresponding to the `Mott insulator to superfluid' transition, and saturate at unity only as the lattice is completely smoothed out.

arXiv:1301.4672•Journal of Physics B Atomic Molecular and Optical Physics•Oct 15, 2013•7 citations
An efficient representation of occupation states of particles on lattices is a critical tool for the exact diagonalization of bosonic Hamiltonians. The memory demands of the traditional method of representing such systems, as arrays of integers, increases rapidly as system size grows, limiting current studies to approximately 16 particles at unit filling. Representing basis vectors using the stars and bars method of combinatorics allows each basis state to be stored as a single 64 bit integer. This optimally compact representation will enable the analysis of new properties of larger bosonic Hamiltonians, including accessible entanglement, which may be useful in evaluating many-body phases as potential candidate quantum resource states.
We introduce a unary coding of bosonic occupation states based on the famous balls and walls counting for the number of configurations of NNN indistinguishable particles on LLL distinguishable sites. Each state is represented by an integer with a human readable bit string that has a compositional structure allowing for the efficient application of operators that locally modify the number of bosons. By exploiting translational and inversion symmetries, we identify a speedup factor of order LLL over current methods when generating the basis states of bosonic lattice models. The unary coding is applied to a one-dimensional Bose-Hubbard Hamiltonian with up to L=N=20L=N=20L=N=20, and the time needed to generate the ground-state block is reduced to a fraction of the diagonalization time. For the ground state symmetry resolved entanglement, we demonstrate that variational approaches restricting the local bosonic Hilbert space could result in an error that scales with system size.
In any bosonic lattice system, which is not dominated by local interactions and thus ``frozen'' in a Mott-type state, numerical methods have to cope with the infinite size of the corresponding Hilbert space even for finite lattice sizes. While it is common practice to restrict the local occupation number basis to Nc{N}_{c}Nc​ lowest occupied states, the presence of a finite condensate fraction requires the complete number basis for an exact representation of the many-body ground state. In this work we present a truncation scheme to account for contributions from higher number states. By simply adding a single coherent-tail state to this common truncation, we demonstrate increased numerical accuracy and the possible increase in numerical efficiency of this method for the Gutzwiller variational wave function and within dynamical mean-field theory.
We demonstrated in the previous chapter (Chapter 1) that the evaluation of matrix elements between states represented by the Slater determinant can be evaluated unambiguously using Slater’s rule [1, 28, 40]. While the determinantal representation and the evaluation matrix elements between the states are quite straightforward, this procedure can eventually be ineffective and cumbersome with increasing size of the system. The occupation number formalism or second quantization technique introduced by Dirac in quantum field theory, in this context has been found to be quite formidable and elegant. This formalism not only allows us to represent the states and the operators in a compact and convenient form but also enables us to deal with systems containing variable numbers of particles without explicitly specifying the number of particles. It implicitly assumes the existence of an unspecified number of functions in a single-particle basis. These one-particle functions, which constitute the Hilbert space for N particles, are generally chosen to be spinorbital because of their orthogonality. In the case of fermions, the many-particle basis is the anti-symmetric product of N one-particle basis functions (Slater determinant). The vector space consisting of all possible anti-symmetric Hilbert spaces is referred to as the Fock space, in which the operators and functions are represented without specifying the number of particles.
To perform efficient many-body calculations in the framework of the exact diagonalization of the Hamiltonian one needs an appropriately tailored Fock basis built from the single-particle orbitals.The simplest way to compose the basis is to choose a finite set of single-particle wave functions and find all possible distributions of a given number of particles in these states.It is known, however, that this construction leads to very inaccurate results since it does not take into account different many-body states having the same energy on equal footing.Here we present a fast and surprisingly simple algorithm for generating the many-body Fock basis built from many-body Fock states having the lowest non-interacting energies.The algorithm is insensitive to details of the distribution of single-particle energies and it can be used for an arbitrary number of particles obeying bosonic or fermionic statistics.Moreover, it can be easily generalized to a larger number of components.Taking as a simple example the system of two ultra-cold bosons in an anharmonic trap, we show that exact calculations in the basis generated with the algorithm are substantially more accurate than calculations performed within the standard approach.
The complexity of quantum many-body problems scales exponentially with the size of the system, rendering any finite-size scaling analysis a formidable challenge. This is particularly true for methods based on the full representation of the wave function, where one simply accepts the enormous Hilbert space dimensions and performs linear algebra operations, e.g., for finding the ground state of the Hamiltonian. If the system satisfies an underlying symmetry where an operator with a degenerate spectrum commutes with the Hamiltonian, it can be block-diagonalized, thus reducing the complexity at the expense of additional bookkeeping. At the most basic level required for Krylov space techniques (like the Lanczos algorithm), it is necessary to implement a matrix-vector product of a block of the Hamiltonian with arbitrary block-wavefunctions, potentially without holding the Hamiltonian block in memory. An efficient implementation of this operation requires the calculation of the position of an arbitrary basis vector in the canonical ordering of the basis of the block. We present here an elegant and powerful, multi-dimensional approach to this problem for the U(1) U ( 1 ) symmetry appearing in problems with particle number conservation. Our divide-and-conquer algorithm uses multiple subsystems and hence generalizes previous approaches to make them scalable. In addition to the theoretical presentation of our algorithm, we provide DanceQ, a flexible and modern – header only – C++20 implementation to manipulate, enumerate, and map to its index any basis state in a given particle number sector as open source software under https://DanceQ.gitlab.io/danceq.
Properties of bosonic atoms in small systems with a periodic quasi one-dimensional circular toroidal lattice potential subjected to rotation are examined by performing exact diagonalization in a truncated many body space. The expansion of the many-body Hamiltonian is considered in terms of the first band Bloch functions, and no assumption regarding restriction to nearest-neighbor hopping (tight-binding approximation) is involved. A finite size version of the zero temperature phase diagrams of Fisher et al. \cite{Fisher} is obtained and the results, in remarkable quantitative correspondence with the results available for larger systems, discussed. Ground state properties relating to superfluidity are examined in the context of two-fluid phenomenology. The basic tool, consisting of the intrinsic inertia associated with small rotation angular velocities in the lab frame, is used to obtain ground state `superfluid fractions' numerically. They are analytically associated with one-body, uniform solenoidal currents in the case of the adopted geometry. These currents are in general incoherent superpositions of contributions from each eigenstate of the associated reduced one-body densities, with the corresponding occupation numbers as weights. Full coherence occurs therefore only when only one eigenstate is occupied by all bosons. The obtained numerical values for the superfluid fractions remain small throughout the parameter region corresponding to the `Mott insulator to superfluid' transition, and saturate at unity only as the lattice is completely smoothed out.