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Efficient Construction of Many-Body Fock States Having the Lowest Energies

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Efficient Construction of Many-Body Fock States Having the Lowest Energies Vol. 136 (2019) ACTA PHYSICA POLONICA A No. 3 Efficient Construction of Many-Body Fock States Having the Lowest Energies A. Chrostowski and T. Sowiński∗ Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland (Received August 7, 2019; revised version 22 September, 2019; in final form September 23, 2019) To perform efficient many-body calculations in the framework of the exact diagonalization of the Hamilto- nian 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. DOI: 10.12693/APhysPolA.136.566 PACS/topics: Fock base generator, exact diagonalization 1. Introduction their derivatives. In the following, we assume that the single-particle part of the Hamiltonian is already diago- The exact diagonalization of the many-body Hamil- nalized and all its eigenstate 'k(r) and their eigenener- tonian [1–3] is one of the simplest and straightforward gies "k (sorted along ascending order) are known. Hav- methods of finding the ground state of the system of in- ing these states, it is very convenient to consider many- teracting quantum particles. It relies on a simple obser- body Hilbert space of N particles as spanned by the Fock vation that having defined a finite set of D Fock states jii states built from these single-particle orbitals. It means (i 2 1;:::; D) one can calculate all matrix elements of that any Fock state can be written formally in the sec- ond quantization formalism as a sequence of numbers of the Hamiltonian H^ of the system, Hij = hijH^ jji, and numerically diagonalize corresponding matrix. As a re- particles occupying individual orbitals sult one obtains approximate decomposition of system’s jii = jn0n1 :::i; (2) P P eigenstates into defined Fock basis jΨii = i αijii and with k nk = N. Of course, in the case of fermionic their eigenenergies Ei. Obviously, the accuracy of this particles, an additional constraint has to be imposed as- straightforward method strongly depends not only on the suring that for any k occupation nk 2 f0; 1g. In this way, number of the Fock states used for calculations but also quantum indistinguishability and statistics are taken into on a particular method the states are chosen from the account. The non-interacting energy of this state is P infinite set of all possible Fock states. Ei = k nk"k. This notation immediately suggests one In typical physical scenario the total Hamiltonian of of the simplest methods of limiting the size of the Fock the many-body system can be divided to the sum of space and choosing only D of them as required by any single-particle Hamiltonians describing dynamics of par- numerical method based on diagonalization [4]. Namely, ticles confined in an external potential V (r) and the re- one limits the number of single-particle orbitals to some maining part describing mutual interactions between par- chosen cut-off C, k 2 f0;:::; Cg, and use them to build all ticles, i.e., the many-body Hamiltonian of N particles has possible Fock states for further calculations. In this ap- a form proach, to increase the numerical accuracy of the many- N 2 2 body diagonalization, one slightly modifies a shape of X ~ @ H^ = − + V (ri) + H^int: (1) single-particle orbitals [5] or increases the cut-off C. Un- 2m @r2 i=1 i fortunately, along with increasing cut-off C the number of Here we do not specify what the form of the interaction Fock states grows exponentially (D = (C + N)!=(C!N!) ^ C Hamiltonian Hint is. In the most general case, it nontriv- and DC = (C + 1)!=[(C − N + 1)!N!] for bosons and ially depends on all particles’ positions fr1;:::; rN g and fermions, respectively). In consequence, the method is naturally limited by numerical resources (mainly by avail- able memory and computational time) and it is com- monly viewed as very demanding from a computational ∗corresponding author; e-mail: [email protected] perspective. (566) Efficient Construction of Many-Body Fock States Having the Lowest Energies 567 To limit exponential growth of the considered Hilbert 2. The algorithm space one can change the way the Fock basis is con- structed. As discussed in the literature (see for exam- Counterintuitively, the best way to present the algo- ple [6–8]), generating the Fock basis directly from the rithm for generating the cropped Fock basis of states limited number of single-particle states is not the most with the lowest non-interacting energy is to represent the efficient way of obtaining well-converged results, since in Fock states in the first quantization formalism, i.e., in- that case an energetic hierarchy in the Fock basis is com- stead of the number of particles occupying single-particle pletely neglected. Namely, this approach takes into ac- orbitals one needs to remember single-particle states oc- count the Fock states with relatively large energies fitting cupied by individual particles. It means that the Fock to small cut-offs C and neglects at the same time other state jii is represented by a set of N numbers (i1; : : : ; iN ) Fock states with substantially lower energies but having rather than a vector (2) encoding occupations. The larger single-particle excitations. As argued in [8], to in- relation between the two is straightforward. As an crease the accuracy of the exact diagonalization without example, the non-interacting ground-state of N = 4 extending numerical efforts, one should select Fock states bosons in both notations reads j1i = j40 :::i = (0; 0; 0; 0) P while in the case of N = 4 fermions it has a form with the lowest non-interacting energy Ei = k nk"k rather than states with the lowest excitations k ≤ C. j1i = j11110 :::i = (0; 1; 2; 3). Since particles are indistin- In general, selecting the Fock states fjiig having the guishable, it is understood that the states encoded in the lowest non-interacting energy is not a trivial task. Only first quantization are appropriately (anti)symmetrized in the case of equally distributed single-particle energies and only for simplicity they are represented by sets with ascending numbers, i ≤ i ≤ ::: ≤ i . "k − "k−1 = ∆ (like in the one-dimensional parabolic 1 2 N confinement) it can be done quite easily since then the The idea of the algorithm is very similar for both quan- non-interacting energies can be represented uniquely by tum statistics. Therefore, let us first perform a short pre- integer numbers. Consequently, the problem can be re- sentation for the case of N bosons. Schematic flowchart duced to the task of finding different partitions of an of the algorithm (in the Fortran-like code) is presented in integer (representing the non-interacting energy) into Fig. 1. The input requirements are: number of particles a fixed number of parts (representing individual parti- N, the single-particle energies "k sorted along ascend- cles) [9]. A detailed explanation is presented for exam- ing order (stored in E(k)), and the maximal energy E ple in [8]. Unfortunately, a simple generalization of this (stored in EnergyMax). As an output one gets the se- approach to cases when single-particle energies are not ries of Fock states fjiig represented by a set of numbers equally distant does not exist. Therefore, one needs to (i1; : : : ; iN ) (stored temporarily in State(j)) which are apply a direct selection of states with the lowest non- generated in lexicographical order. Example outputs of interacting energy from a basis of states having the low- the algorithm for two different situations are presented in est excitations (determined by a single-particle cut-off Table I. Table Ia presents resulting Fock basis for the one- C). This, however, is very inefficient and consumes huge dimensional harmonic confinement having equally dis- amounts of computational facilities (memory and com- tant single-particle energy levels putation time). Consequently, its usefulness is strongly "0 = 0:5; (3a) limited. " = " + 1; for k > 0 (3b) In this work, we present a simple numerical algorithm k k−1 with the maximal energy E = 6. As it is seen, only of generating the cropped Fock basis for N particles twelve Fock states have this property (D = 12). At (bosons as well as fermions) in the general case of any E this point, one should note a huge reduction of the con- complete set of single-particle orbitals ' (r) having as- k sidered Hilbert space when compared to the standard cending single-particle energies " ≤ " ≤ :::. As the in- 0 1 approach of single-particle cut-off. It is clearly seen that put, the algorithm requires only one parameter, namely in the case of the former method, to capture all states the maximal energy E of states in generated Fock ba- with non-interacting energy no larger than E = 6, one sis.
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