In this work we build the foundations of a quantum Monte Carlo as a stochastic numerical method to solve lattice many-body quantum systems with nearest-neighbor interactions at most. As motivation, we briefly describe the bilinear-biquadratic Heisenberg model with an external field, for spin-1 particles, as an effective Hamiltonian of the Bose-Hubbard model with an external quadratic Zeeman field in the Mott insulator phase at unit filling. Then, we discuss how to implement the world line Monte Carlo with local updates to circumvent the difficulties that arise on these type of systems by mapping the quantum partition function into the one of an effective classical model, in one additional dimension, given by the imaginary time evolution of the system. Such a mapping is performed by means of the Suzuki-Trotter decomposition, which transforms the original partition function into a summation of weights given by the classical configurations. Later, we present a set of observables that can be measured through this method and show how to use a Metropolis update scheme to accomplish the measurements. At last, we present the maximization of the configuration weights for three parameter sets as the first and relevant step to perform future measurements.
In this work we build the foundations of a quantum Monte Carlo as a stochastic numerical method to solve lattice many-body quantum systems with nearest-neighbor interactions at most. As motivation, we briefly describe the bilinear-biquadratic Heisenberg model with an external field, for spin-1 particles, as an effective Hamiltonian of the Bose-Hubbard model with an external quadratic Zeeman field in the Mott insulator phase at unit filling. Then, we discuss how to implement the world line Monte Carlo with local updates to circumvent the difficulties that arise on these type of systems by mapping the quantum partition function into the one of an effective classical model, in one additional dimension, given by the imaginary time evolution of the system. Such a mapping is performed by means of the Suzuki-Trotter decomposition, which transforms the original partition function into a summation of weights given by the classical configurations. Later, we present a set of observables that can be measured through this method and show how to use a Metropolis update scheme to accomplish the measurements. At last, we present the maximization of the configuration weights for three parameter sets as the first and relevant step to perform future measurements.
The World-Line Quantum Monte Carlo method is a stochastic numerical analysis tool, the implementation of which is of particular relevance in the study of quantum many-body systems. These systems are often difficult to study using analytical techniques or conventional numerical methods. The objective of this study is to simulate a reticular bosonic spin 1 system in the Mott insulator phase, with the aim of characterizing the quantum phase transition between the XY-Ferro and Large D magnetic phases by means of the World-Lines Quantum Monte Carlo. The study highlights the importance of characterizing the magnetic phases using local and non-local observables, which provide information about the magnetic properties of the system in the z-axis. In addition, the transition between the Large D and XY-Ferro phases is analyzed and compared with the results obtained by the MPS algorithm and the field theory. The document begins by establishing the physics of the Bose-Hubbard model, which is a theoretical model for an ultracold system of bosonic spin-1 particles in an optical lattice. The inclusion of the spin degree of freedom in the model gives rise to two types of interactions between the bosons: one that preserves the spin projection after the interaction and one that modifies it. To study the magnetic properties of the system, a quadratic Zeeman field is incorporated into the model, which partially lifts the degeneracy and minimizes the energy of the system depending on the field strength and the magnetic projections of the bosons. Subsequently, Quasi-Degenerate Perturbation Theory is applied, considering the particles’ hopping between lattice sites as the perturbative term, to obtain an effective model of the system. The Bilinear-Biquadratic Heisenberg model is obtained, which restricts the system to the Mott insulator phase and is parametrized by the boson’s interaction strength θ and the field strength D. Finally, the condition of a fixed number of bosons in the lattice is imposed on the canonical partition function of the system. After performing algebraic manipulations to facilitate its calculation, the functional form of a partition function for a grand canonical ensemble in the presence of a linear Zeeman field is retrieved. The algorithm implementation involves the transformation of the partition function of the quantum system to another partition function corresponding to that of a classical system with an additional dimension, the imaginary time, which stores the information about the thermalization of the system ground-state. The Metropolis algorithm is addressed, this method implements the importance sampling and allows the measurement of local and non-local observables through estimators. Specifically, it is possible to measure the internal energy, the magnetization, the total number of bosons, the quadrupole density and two types of spin correlations. In addition, the magnetic susceptibility of the system was measured, which allows the construction of the phase diagram for the transition between the XY-Ferro and Large D phases.
Abstract By precisely writing down the matrix element of the local Boltzmann operator (e-th, where h is the Hermitian conjugate pairs of off-diagonal operators), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. With current formula, the Hubbard–Stratonovich transformation is not necessary, and is not based on the determinant approach, which can improve the computational efficiency. The results show that, the simulation time has the square-law scaling with system sizes, which is comparable with the usual first-principle calculation. The current formula also improves the accuracy of the Suzuki–Trotter decomposition. As an example, we have studied the one-dimensional half-filled Hubbard model at finite temperature. The obtained results are in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm can be used in various studies.
By precisely writing down the matrix element of the local Boltzmann operator ([Formula: see text], where [Formula: see text] is the Hermitian conjugate pairs of off-diagonal operators), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. With the current formula, the Hubbard-Stratonovich transformation is not necessary, accordingly the determinant calculation is not needed, which can improve the computational efficiency. The results show that, the simulation time has the square-law scaling with system sizes, which is comparable with the usual first-principles calculations. The current formula also improves the accuracy of the Suzuki-Trotter decomposition. As an example, we have studied the one-dimensional half-filled Hubbard model at finite temperature. The obtained results are in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm could be applied to various studies in future.
By precisely writing down the matrix element of the local Boltzmann operator, we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. With current formula, the Hubbard-Stratonovich transformation is not necessary, and is not based on the determinant approach, which can improve the computational efficiency. The results show that, the simulation time has the square-law scaling with system sizes, which is comparable with the usual first-principle calculation. The current formula also improves the accuracy of the Suzuki-Trotter decomposition. As an example, we have studied the one-dimensional half-filled Hubbard model at finite temperature. The obtained results are in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm can be used in various studies.
By precisely writing down the matrix element of the local Boltzmann operator (e(−τh), where h is the paired operator formed by two interacting terms conjugate to each other), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. Our formula presents remarkable outcomes such that: 1) the Hubbard-Stratonovich transformation is not necessary, and the auxiliary field is no longer needed, which can improve the computational efficiency. 2) The new formula improves the accuracy of the Suzuki-Trotter decomposition. Our method also provides a potential theoretical framework for analyzing and solving the fermion sign problem. As an example, we studied the one-dimensional half-filled Hubbard model at a finite temperature. The obtained results were in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm could be used in various studies.
Simulating long-range interacting systems is a challenging task due to its computational complexity that the computational effort for each local update is of order \mathcal{O}(N) 𝒪 ( N ) , where N N is the size of the system. In this work, we introduce the clock factorized quantum Monte Carlo method, an efficient technique for simulating long-range interacting quantum systems. The method is developed by generalizing the clock Monte Carlo method for classical systems [Phys. Rev. E 99 010105 (2019)] to the path-integral representation of long-range interacting quantum systems, with some specific treatments for quantum cases and a few significant technical improvements in general. We first explain how the clock factorized quantum Monte Carlo method is implemented to reduce the computational overhead from \mathcal{O}(N) 𝒪 ( N ) to \mathcal{O}(1) 𝒪 ( 1 ) . In particular, the core ingredients, including the concepts of bound probabilities and bound rejection events, the recursive sampling procedure, and the fast algorithms for sampling an extensive set of discrete and small probabilities, are elaborated. Next, we show how the clock factorized quantum Monte Carlo method can be flexibly implemented in various update strategies, like the Metropolis and worm-type algorithms. Finally, we demonstrate the high efficiency of the clock factorized quantum Monte Carlo algorithms using examples of three typical long-range interacting quantum systems, including the transverse field Ising model with long-range z z - z z interaction, the extended Bose-Hubbard model with long-range density-density interactions, and the XXZ Heisenberg model with long-range spin interactions. We expect that the clock factorized quantum Monte Carlo method would find broad applications in statistical and condensed-matter physics.