CC BY
6
i l St. Petersburg Polytechnic University Journal. Physics and Mathematics. 2022 Vol. 15, No. 3.1 Научно-технические ведомости СПбГПУ. Физико-математические науки. 15 (3.1) 2022
Conference materials UDC 536.91
DOI: https://doi.org/10.18721/JPM.153.114
Hybrid Monte Carlo algorithm for studying the Edwards-Anderson model
A. E. Rybin ', 2 D. Yu. Kapitan 1 2, K. V. Nefedev 1 2, A. G. Makarov 1 2, V. Yu. Kapitan 1 2 1 Far Eastern Federal University, Vladivostok, Russia;
2 Institute of Applied Mathematics of Far Eastern Branch, RAS, Vladivostok, Russia
H rybin.ae@dvfu.ru
Abstract: The complexity of the study of spin glasses is related to their frustrations, due to which classical Monte Carlo algorithms experience serious difficulties when trying to calculate such systems. The main object of research in this paper is two-dimensional Edwards-Anderson model on a square lattice. In the paper, we propose an optimized Hybrid Monte Carlo method for calculating the values of thermodynamic averages and ground state energies of the frustrated spin glass model. The validity of the results is confirmed by comparison with numerical simulation with the parallel tempering Monte Carlo method, complete enumeration algorithm and robust machine learning approach-RBM neural network. The proposed algorithm has a number of advantages: possible high parallelization of the algorithm to speed up simulation, calculation accuracy and low resource consumption, which allows to calculate lattices of relatively large size. This algorithm can be applied to calculations of lattices with different geometry and sizes.
Keywords: Edwards-Anderson model, Monte Carlo algorithm, ground state, frustration
Funding: Research on hybrid algorithm and development of the numerical calculation algorithm as a whole was carried out with the financial support of the grant of the President of the Russian Federation for state support of leading scientific schools of the Russian Federation (NSh-2559.2022.1.2). Computer program development and coding of parallel efficient code were carried out within the framework of State Task of the Ministry of Science and Higher Education of the Russian Federation (№ 0657-2020-0005).
Citation: Rybin A. E., Kapitan D. Yu., Nefedev K. V., Makarov A. G., Kapitan V. Yu. Hybrid Monte Carlo algorithm for studying the Edwards-Anderson model. St. Petersburg State Polytechnical University Journal. Physics and Mathematics. 15 (3.1) (2022) 82—86. DOI: https://doi.org/10.18721/JPM.153.114
This is an open access article under the CC BY-NC 4.0 license (https://creativecommons. org/licenses/by-nc/4.0/)
Материалы конференции УДК 536.91
DOI: https://doi.org/10.18721/JPM.153.114
Гибридный алгоритм Монте-Карло для изучения модели Эдвардса-Андерсона
А. Е. Рыбин ', 2 и, Д. Ю. Капитан 1 2, К. В. Нефедев 1 2, А. Г. Макаров 1 2, В. Ю. Капитан 1 2 1 Дальневосточный Федеральный Университет, г. Владивосток, Россия; 2 Институт прикладной математики Дальневосточного отделения РАН, г. Владивосток, Россия
и rybin.ae@dvfu.ru
Аннотация. Сложность исследования спиновых стекол связана с фрустрациями, из-за которых классические алгоритмы Монте-Карло испытывают серьезные трудности
© Rybin A. E., Kapitan D. Yu., Nefedev K. V., Makarov A. G., Kapitan V. Yu., 2022. Published by Peter the Great St. Petersburg Polytechnic University.
при попытке расчета таких систем. Основным объектом исследования в данной работе является двумерная модель Эдвардса-Андерсона на квадратной решетке. В работе предложен оптимизированный гибридный метод Монте-Карло для расчета значений термодинамических средних и энергий основного состояния модели фрустрированного спинового стекла. Достоверность результатов подтверждается сравнением с численным моделированием с помощью параллельного отжига методом Монте-Карло, алгоритмом полного перебора и надежным алгоритмом машинного обучения - нейронной сети RBM. Предложенный алгоритм обладает рядом преимуществ: возможность параллелизации алгоритма для ускорения моделирования, точность расчетов и низкое потребление ресурсов, что позволяет рассчитывать решетки относительно большого размера. Данный алгоритм может быть применен для расчета решеток с различной геометрией и размерами.
Ключевые слова: модель Эдвардса-Андерсона, алгоритм Монте-Карло, основное состояние, фрустрации
Финансирование: Исследование гибридного алгоритма и разработка алгоритма численного расчета в целом проводились при финансовой поддержке гранта Президента Российской Федерации для государственной поддержки ведущих научных школ Российской Федерации (НШ-2559.2022.1.2). Разработка компьютерной программы и кодирование эффективного параллельного кода проводились в рамках государственного задания Министерства науки и высшего образования Российской Федерации (№ 0657-2020-0005).
Ссылка при цитировании: Рыбин А. Е., Капитан Д. Ю., Нефедев К. В., Макаров А. Г., Капитан В. Ю., Гибридный алгоритм Монте-Карло для изучения модели Эдвардса-Андерсона // Научно-технические ведомости СПбГПУ. Физико-математические науки. 2022. Т. 15. № 3.1. С. 82-86. DOI: https://doi.org/10.18721/ JPM.153.114
Статья открытого доступа, распространяемая по лицензии CC BY-NC 4.0 (https:// creativecommons.org/licenses/by-nc/4.0/)
Introduction
Nowadays, frustrated spin systems, such as spin glasses and ices, are actively studied in modern condensed matter physics [1-3]. In such models, frustrations arise due to the special lattice topology or the competition of exchange interactions, which leads to the search for new magnetic ground states and unique effects that can arise.
Spin glasses are disordered magnetics, which are characterized by two unique properties that greatly distinguish these systems from others: in such systems there is strong competition between ferromagnetic and antiferromagnetic interactions, i.e., frustrations, and disorder, which is freezing (or solidifying) of atoms in different locations during alloy formation. Many processes occurring in spin glasses cannot be described within the framework of the classical theory of phase transitions and require some new non-standard approaches.
The process of reaching the ground states and studying the thermodynamic properties even in the simplest model, the Ising model, is associated with serious difficulties in theoretical analysis and numerical calculations. This problem is much more complicated than it seems at first sight. Creating an efficient algorithm for finding the ground states is one of the main challenges in the theory of frustrated magnetism.
The classical Monte Carlo algorithm and its many optimizations have their drawbacks. Singlespin sampling methods near critical points drastically lose in efficiency, the so-called 'critical slowdown' occurs. In addition, at very low temperatures, only a small fraction of spins makes a flip. The motion of the system in phase space is very slow, so the system requires generation of a very large number of states to move to an equilibrium state. In turn, the use of multicanonical methods has difficulties in calculating the thermodynamics of systems with a relatively large number of spins.
To solve the problem of searching for thermodynamics of frustrated models of many interacting bodies, as well as to search for ground state configurations, a new hybrid multispin method is applied, which is a logical continuation of the Metropolis algorithm.
© Рыбин А. Е., Капитан Д. Ю., Нефедев К. В., Макаров А. Г., Капитан В. Ю., 2022. Издатель: Санкт-Петербургский политехнический университет Петра Великого.
Edwards-Anderson model
The main object of research in this paper is two-dimensional Edwards-Anderson model on a square lattice (see Fig. 1). Edwards and Anderson proposed in [4] to change the distribution function of the exchange interaction to a more complex one, where the exchange integral J is a random function and the average value of J. is equal to zero. In such systems, half of the spins has ferromagnetic interaction with its nearest neighbors, and the other part is antiferromagnetic one.
The interaction J between the spin pair (ij) changes during the transition from one pair to another. The Hamiltonian is then expressed as:
H =-E J^SSj - S, (1)
i, j i
S , S. are the spins in lattice, {ij) denotes summation over pairs of interacting spins in a system with the size N, h is the external magnetic field. Edwards and Anderson suggested in [4] that each J., distributed independently according to the probability distribution P(Jt). In this work, we have used the bimodal distribution, i.e., the amount of positive and negative bond values is equal.
The average energy of the spin glass model at temperature T is calculated by Eq. (2) and the average magnetization by Eq. (3).
£ (t) = -1{ ■ <2)
(3)
Based on these two main characteristics, we can calculate the various thermodynamic properties, such as magnetic susceptibility (4) and specific heat (5)
C(T) = ■ (5)
v ' N kBT2
In this study, we investigated models on the square lattice with different number of spins, such as N = 6x6, 10x10, 20x20, 30x30.
Parallel Hybrid Monte-Carlo Algorithm:
Initially, the hybrid algorithm was proposed for field theory studies on a lattice with fermionic degrees of freedom [5]. Subsequently, this algorithm was transferred and adapted for various problems [6, 7]. More recently, we proposed an approach for studying the temperature behavior of the frustration parameter in a geometrically frustrated hexagonal lattice [8]. It is based on combining the canonical and multicanonical sampling of the Gibbs distribution.
However, after comparing the results of the algorithm with the complete enumeration method, it turned out that as the size of the systems increased, the algorithm did not arrive at the ground state. For this reason, the algorithm was optimized and reduced to the form presented below.
• ^ random spins and bonds initialization
• ^ for all threads do
o ^ choosing random spin in the system if cluster borders are not crossing then
■ ^ copy cluster spin array to private memory
■ ^ for 2*Nc steps do
• ^ for every spin in the cluster do
o ^ current spin value = ((counter >> 2* Nc —1) & 1) o ^ if new energy<previous energy then
■ ^ save current configuration as minimal configuration o ^collect spin values in main system array
• ^ ground state and thermodynamics calculation.
Results and Discussion
To demonstrate the work of the algorithm, we compare its accuracy with different methods [9—11], such as complete enumeration algorithm, ML approach using RBM, see Table 1. Algorithms were tested on a square lattice of the two-dimensional Edwards-Anderson model, with number of spins N = 6*6, 10x10, 20*20, 30*30. As can be seen from the table, proposed hybrid Monte Carlo algorithm demonstrates the same accuracy as the complete enumeration algorithm, but, unlike the latter, it is much less resource-intensive, which allows us to calculate larger systems.
Table 1
Comparison of results obtained by different computational approaches
System size E . min
CE HM RBM
6 x 6 -1.30 -1.30 -1.30
10 x 10 - -1.40 -1.40
20 x 20 - -1.38 -1.38
30 x 30 - -1.34 -1.34
In contrast to the other algorithms presented in Table 1, the hybrid Monte Carlo algorithm can be used to study other important properties of the system, such as the dependence of the staggered magnetization and the system degeneracy multiplicity on the external magnetic field. It also allows the calculation of basic thermodynamic quantities such as heat capacity and magnetic susceptibility. A comparative graph of the dependence of heat capacity on temperature is shown in Fig. 1. We checked the accuracy of the proposed algorithm by comparing it with the parallel tempering (PT) method, which has proved [12—14] to be
Fig. 1. Dependence of heat capacity on temperature according to Edwards-Anderson with N = 20*20, obtained by Hybrid Monte Carlo (HMC) and Parallel Tempering (PT) algorithms
Based on the values obtained, we can conclude that the optimized hybrid Monte Carlo algorithm is well adapted to searching for the main states of complex lattices, as well as to the calculation of the main thermodynamic characteristics.
Conclusion
In this work, we proposed and implemented an optimization of a hybrid Monte Carlo algorithm. Using this method, various important characteristics on spin glass lattices with different numbers of spins were obtained, such as: energy of ground states, dependence of heat capacity and susceptibility on temperature. The validity of the obtained results was validated using well-proven algorithms. Due to its accuracy and versatility, this algorithm can be used to study systems with a large number of spins and different lattice geometries.
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THE AUTHORS
RYBIN Alexey E.
rybin.ae@dvfu.ru
ORCID: 0000-0002-1055-9217
makarov.ag@dvfu.ru ORCID: 0000-0002-7778-8364
MAKAROV Aleksandr G.
KAPITAN Dmitrii Yu.
kapitan.diu@dvfu.ru ORCID: 0000-0001-9815-1891
KAPITAN Vitalii Yu.
kapitan.vyu@dvfu.ru ORCID: 0000-0002-5068-8910
NEFEDEV Konstantin V.
nefedev.kv@dvfu.ru ORCID: 0000-0001-7330-5137
Received 22.05.2022. Approved after reviewing 18.07.2022. Accepted 19.07.2022.
© Peter the Great St. Petersburg Polytechnic University, 2022
