H Φ [ aitch-phi ] is a program package based on the Lanczos-type eigenvalue solution applicable to a broad range of quantum lattice models, i.e., arbitrary quantum lattice models with two-body interactions, including the Heisenberg model, the Kitaev model, the Hubbard model and the Kondo-lattice model. While it works well on PCs and PC-clusters, H Φ also runs efficiently on massively parallel computers, which considerably extends the tractable range of the system size. In addition, unlike most existing packages, H Φ supports finite-temperature calculations through the method of thermal pure quantum (TPQ) states. In this paper, we explain theoretical background and user-interface of H Φ . We also show the benchmark results of H Φ on supercomputers such as the K computer at RIKEN Advanced Institute for Computational Science (AICS) and SGI ICE XA (Sekirei) at the Institute for the Solid State Physics (ISSP). Program summary Program Title: H Φ Program Files doi: http://dx.doi.org/10.17632/vnfthtyctm.1 Licensing provisions: GNU General Public License, version 3 or later Programming language: C External routines/libraries: MPI, BLAS, LAPACK Nature of problem: Physical properties (such as the magnetic moment, the specific heat, the spin susceptibility) of strongly correlated electrons at zero- and finite-temperature. Solution method: Application software based on the full diagonalization method, the exact diagonalization using the Lanczos method, and the microcanonical thermal pure quantum state for quantum lattice model such as the Hubbard model, the Heisenberg model and the Kondo model. Restrictions: System size less than about 20 sites for an itinerant electronic system and 40 site for a local spin system. Unusual features: Finite-temperature calculation of the strongly correlated electronic system based on the iterative scheme to construct the thermal pure quantum state. This method is efficient for highly frustrated system which is difficult to treat with other methods such as the unbiased quantum Monte Carlo.
H Φ [ aitch-phi ] is a program package based on the Lanczos-type eigenvalue solution applicable to a broad range of quantum lattice models, i.e., arbitrary quantum lattice models with two-body interactions, including the Heisenberg model, the Kitaev model, the Hubbard model and the Kondo-lattice model. While it works well on PCs and PC-clusters, H Φ also runs efficiently on massively parallel computers, which considerably extends the tractable range of the system size. In addition, unlike most existing packages, H Φ supports finite-temperature calculations through the method of thermal pure quantum (TPQ) states. In this paper, we explain theoretical background and user-interface of H Φ . We also show the benchmark results of H Φ on supercomputers such as the K computer at RIKEN Advanced Institute for Computational Science (AICS) and SGI ICE XA (Sekirei) at the Institute for the Solid State Physics (ISSP). Program summary Program Title: H Φ Program Files doi: http://dx.doi.org/10.17632/vnfthtyctm.1 Licensing provisions: GNU General Public License, version 3 or later Programming language: C External routines/libraries: MPI, BLAS, LAPACK Nature of problem: Physical properties (such as the magnetic moment, the specific heat, the spin susceptibility) of strongly correlated electrons at zero- and finite-temperature. Solution method: Application software based on the full diagonalization method, the exact diagonalization using the Lanczos method, and the microcanonical thermal pure quantum state for quantum lattice model such as the Hubbard model, the Heisenberg model and the Kondo model. Restrictions: System size less than about 20 sites for an itinerant electronic system and 40 site for a local spin system. Unusual features: Finite-temperature calculation of the strongly correlated electronic system based on the iterative scheme to construct the thermal pure quantum state. This method is efficient for highly frustrated system which is difficult to treat with other methods such as the unbiased quantum Monte Carlo.
HΦ [aitch-phi] is an open-source software package of numerically exact and stochastic calculations for a wide range of quantum many-body systems. In this paper, we present the newly added functions and the implemented methods in vers. 2 and 3. In ver. 2, we implement spectrum calculations by the shifted Krylov method, and low-energy excited state calculations by the locally optimal blocking preconditioned conjugate gradient (LOBPCG) method. In ver. 3, we implement the full diagonalization method using ScaLAPACK and GPGPU computing via MAGMA. We also implement a real-time evolution method and the canonical thermal pure quantum (cTPQ) state method for finite-temperature calculations. The Wannier90 format for specifying the Hamiltonians is also implemented. Using the Wannier90 format, it is possible to perform the calculations for the ab initio low-energy effective Hamiltonians of solids obtained by the open-source software RESPACK. We also update Standard mode—simplified input format in HΦ—to use these functions and methods. We explain the basics of the implemented methods and how to use them. Program Title: HΦ [aitch-phi] CPC Library link to program files: https://doi.org/10.17632/vnfthtyctm.2 Developer's repository link: https://github.com/issp-center-dev/HPhi Licensing provisions: GPLv3 Programming language: C, Fortran Journal reference of previous version: Comput. Phys. Commun. 217 (2017) 180–192 Does the new version supersede the previous version?: Yes. The latest version has compatibility with the old versions. Although most functions are available in newer versions, some redundant/unnecessary functions were abolished in newer versions. Reasons for the new version: Implementation of new functions and methods, development of utilities, and bug fixes. Summary of revisions: We added new functions to obtain excited spectrum, low-energy excited states, and time-dependent physical quantities. We also implemented the full diagonalization method by using MAGMA on GPGPUs and finite-temperature simulations by using the canonical thermal pure quantum state method. In addition, we developed utilities for connection with RESPACK and a submodule for a generator of input files. Nature of problem: Physical properties in quantum lattice models with finite system sizes Solution method: In HΦ, we implemented several numerical methods such as the Lanczos method, the full diagonalization method, the locally optimal block preconditioned conjugate gradient (LOBPCG) method, the real-time evolution method based on the Taylor expansion, the shifted Krylov method, and the microcanonical/canonical thermal pure quantum state method.
We develop a program package named QS3 based on the (thick-restart) Lanczos method for analyzing spin-1/2 XXZ-type quantum spin models on spatially uniform/non-uniform lattices near fully polarized states, which can be mapped to dilute hardcore Bose systems. All calculations in QS3, including eigenvalue problems, expectation values for one/two-point spin operators, and static/dynamical spin structure factors, are performed in the symmetry-adapted bases specified by the number N↓ of down spins and the wave number k associated with the translational symmetry without using the bit representation for specifying spin configurations. Because of these treatments, QS3 can support large-scale quantum systems containing more than 1000 sites with dilute N↓. We show the benchmark results of QS3 for the low-energy excitation dispersion of the isotropic Heisenberg model on the 10×10×10 cubic lattice, the static and dynamical spin structure factors of the isotropic Heisenberg model on the 10×10 square lattice, and the open-MP parallelization efficiency on the supercomputer (Ohtaka) based on AMD Epyc 7702 installed at the Institute for the Solid State Physics (ISSP). Theoretical backgrounds and the user interface of QS3 are also described. Program Title: QS3 CPC Library link to program files: https://doi.org/10.17632/yc6jytvf8v.1 Developer's repository link: https://github.com/QS-Cube/ED Licensing provisions: MIT Programming language: Fortran90 External routines/libraries: BLAS, LAPACK Nature of problem: Physical properties (such as total energy, magnetic moment, two-point spin correlation function, and dynamical structure factor) Solution method: Application software based on the full diagonalization method and the exact diagonalization method using the Lanczos and thick-restart Lanczos techniques for quantum spin S=1/2 models such as the XXZ model. Restrictions: Spin S=1/2 systems with U(1) symmetry. Unusual features: Massively large quantum spin systems with U(1) symmetry near the saturation field can be solved with or without considering translational symmetry, which is difficult to treat using standard exact diagonalization libraries with the bit representation for specifying spin configurations.
The physics of strongly correlated materials poses one of the most challenging problems in condensed-matter sciences. Standard approximations applicable to wide classes of materials such as the local density approximation fail, due to the importance of the Coulomb repulsion between localized electrons. Instead, we resort to non-perturbative many-body methods. The calculations are, however, only feasible for rather small model systems. The full Hamiltonian of a real material is approximated by a model Hamiltonian comprising only the most important electronic degrees of freedom, while the effect of all other electrons is included in an average way by renormalizing the parameters. Realistic calculations of strongly correlated materials need to include sufficiently many of these electronic degrees of freedom. The new generation of massively parallel supercomputers allows for these realistic calculations. However, exploiting their computational power requires newly devised algorithms. As a solver we use the Lanczos method, which needs the full many-body state of the correlated system. It is thus limited by the available main memory. The foremost problem for a distributed-memory implementation is that the multiplication of the Hamiltonian to the many-body state leads to highly non-local memory access patterns. A simple yet important observation leads to an efficient solution: in the kinetic term of the Hamiltonian the electron-spin is conserved. Thus, writing the many-body vector as a matrix v(iup,idn), where the indices label spin-configurations, we find that the hopping term only connects vector elements that differ in one index. Hence, storing entire slices v(iup,:) on one node, the kinetic term for the spin-down electrons is local to that process. After transposing v, the same is true for the hopping of the spin-up electrons. With an efficient matrix transpose, implemented via MPI collectives, we thus obtain a version of the Lanczos method that efficiently well on JUBL, Jülich's Blue Gene/L. Using the code we treated two-orbital Hubbard Models which show interesting Mott-band insulator transitions. Kohn's criterium which determines the phase of a system by ground-state properties alone, namely the response of the ground-state energies to a vector potential, was employed. This vector potential can be equivalently regarded as a change in boundary conditions, requiring complex wave vectors. We found that half-filled systems become metallic in the region between a band and Mott insulating phase. To extrapolate to infinite system sizes and therefore getting access to an arbitrarily high resolution in reciprocal space, we use cluster perturbation theory (CPT). It works by diagonalizing a finite cluster exactly and then treating the hopping between identical clusters perturbatively in strong coupling perturbation theory. This leads to an effectively infinite chain. Due to our efficient Lanczos implementation it is possible to calculate relatively large clusters, minimizing finite-size effects. The combination of our Lanczos code and CPT allowed for a treatment of the the organic metal TTF-TCNQ. We were able to solve a long-standing problem in the interpretation of its experimental spectra. Being an interesting one-dimensional metal TTF-TCNQ has been studied thoroughly for more than 30 years. Theoretical estimates as well as experimental measurements suggest values for the parameters of the Hubbard model t and U. With these values Hubbard model calculations show qualitatively the same features as APRES experiments, namely the signatures of spin-charge separation at the Gamma-point. But in order to fit the experimental data to the numerical calculations t has to be doubled, which translates into a broadening of the spectra. This ad-hoc adjustment is not only unsatisfactory but also the temperature dependence of the spectral function is incorrectly reproduced. We resolved this problem by properly including nearest-neighbor Coulomb interactions. Recent DFT calculations show that indeed the commonly used values of t and U are correct. But Coulomb interactions between electrons on neighboring sites are also significant and must not be neglected. Thus, we studied the effect of the next-neighbor interaction parameter V being at about U/2. Including V in our Lanczos calculations for obtaining the angular-integrated spectral function shows that its effect is to broaden the spectra, comparable to increasing the value of t. Moreover, we found that doubling t indeed mimics the effect of Vapprox U/2. Surprisingly, the broadening of the spectrum as a result of V can be understood in first-order Rayleigh-Schrödinger perturbation theory.
mVMC (many-variable Variational Monte Carlo) is an open-source software package based on the variational Monte Carlo method applicable for a wide range of Hamiltonians for interacting fermion systems. In mVMC, we introduce more than ten thousands variational parameters and simultaneously optimize them by using the stochastic reconfiguration (SR) method. In this paper, we explain basics and user interfaces of mVMC. By using mVMC, users can perform the calculation by preparing only one input file of about ten lines for widely studied quantum lattice models, and can also perform it for general Hamiltonians by preparing several additional input files. We show the benchmark results of mVMC for the Hubbard model, the Heisenberg model, and the Kondo-lattice model. These benchmark results demonstrate that mVMC provides ground-state and low-energy-excited-state wave functions for interacting fermion systems with high accuracy. Program title: mVMC Program Files doi: http://dx.doi.org/10.17632/xhgyp6ncvt.1 Licensing provisions: GNU General Public License version 3 Programming language: C External routines/libraries: MPI, BLAS, LAPACK, Pfapack, ScaLAPACK (optional) Nature of problem: Physical properties (such as the charge/spin structure factors) of strongly correlated electrons at zero temperature. Solution method: Application software based on the variational Monte Carlo method for quantum lattice model such as the Hubbard model, the Heisenberg model and the Kondo model. Unusual features: It is possible to perform the highly-accurate calculations for ground states in a wide range of theoretical Hamiltonians in quantum many-body systems. In addition to the conventional orders such as magnetic and/or charge orders, user can treat the anisotropic superconductivities within the same framework. This flexibility is the main advantage of mVMC.
Abstract The development of novel quantum many-body computational algorithms relies on robust benchmarking. However, generating such benchmarks is often hindered by the massive computational resources required for exact diagonalization or quantum Monte Carlo simulations, particularly at finite temperatures. In this work, we propose a new algorithm for obtaining thermal pure quantum states, which allows efficient computation of both mechanical and thermodynamic properties at finite temperatures. We implement this algorithm in our open-source C++ template library, Physica . Combining the improved algorithm with state-of-the-art software engineering, our implementation achieves high performance and numerical stability. As an example, we demonstrate that for the 4 × 4 Hubbard model, our method runs approximately 10 3 times faster than ℋ Φ 3.5.2. Moreover, the accessible temperature range is extended down to β = 32 across arbitrary doping levels. These advances significantly push forward the frontiers of benchmarking for quantum many-body systems.
A new computational method for finite-temperature properties of strongly correlated electrons is proposed by extending the variational Monte Carlo method originally developed for the ground state. The method is based on the path integral in the imaginary-time formulation, starting from the infinite-temperature state that is well approximated by a small number of certain random initial states. Lower temperatures are progressively reached by the imaginary-time evolution. The algorithm follows the framework of the quantum transfer matrix and finite-temperature Lanczos methods, but we extends them to treat much larger system sizes without the negative sign problem by optimizing the truncated Hilbert space on the basis of the time-dependent variational principle (TDVP). This optimization algorithm is equivalent to the stochastic reconfiguration (SR) method that has been frequently used for the ground state to optimally truncate the Hilbert space. The obtained finite-temperature states allow an interpretation based on the thermal pure quantum (TPQ) state instead of the conventional canonical-ensemble average. Our method is tested for the one- and two-dimensional Hubbard models and its accuracy and efficiency are demonstrated.