=Paper=
{{Paper
|id=Vol-2930/paper27
|storemode=property
|title=Optimization of the Hybrid Monte-Carlo Algorithm for the Edwards-Anderson Model
|pdfUrl=https://ceur-ws.org/Vol-2930/paper27.pdf
|volume=Vol-2930
|authors=Dmitrii Kapitan,Alexey Rybin,Petr Andriushchenko,Aleksandr Makarov,Yuriy Shevchenko,Konstantin Soldatov,Vitalii Kapitan,Konstantin Nefedev
}}
==Optimization of the Hybrid Monte-Carlo Algorithm for the Edwards-Anderson Model==
https://ceur-ws.org/Vol-2930/paper27.pdf
Optimization of the Hybrid Monte-Carlo Algorithm for the
Edwards-Anderson Model
Dmitrii Yu. Kapitana,b, Alexey E. Rybina,b, Petr D. Andriushchenkoc, Aleksandr G.
Makarova,b, Yuriy. A. Shevchenkoa,b, Konstantin S. Soldatova,b , Vitalii Yu. Kapitana,b and
Konstantin V. Nefedeva,b
a
Department of Computer Systems, School of Natural Sciences, Far Eastern Federal University, Vladivostok,
690922, 10 Ajax Bay, Russian Federation
b
Institute of Applied Mathematics, Far Eastern Branch, Russian Academy of Science, Vladivostok, 690041, 7
Radio St., Russian Federation
c
ITMO University, Kronverksky Pr. 49, bldg. A, St. Petersburg, 197101, Russian Federation
Abstract
This article describes the principle of using Hybrid Monte-Carlo method in spin glasses using
the Edwards-Anderson model as an example. We consider efficient algorithm for searching
ground states of frustrated systems. We discuss two optimizations for this algorithm in order
to find the most efficient. We implement and test algorithm on a two-dimensional square
lattice of Edwards-Anderson model. The advantages of using the Hybrid Monte-Carlo
method in spin glasses are revealed.
Keywords 1
Hybrid Monte-Carlo algorithm, Spin Glass, Ground State, Edwards-Anderson model
1. Introduction
Frustrated magnetic interactions are one of the most fiercely debated topics in condensed matter
physics [1, 2]. Interest in spin systems where frustrations, as a result of a special lattice topology or
competition exchange interactions, suppresses the Neel antiferromagnetic order is greatly stimulated
by the search for new magnetic ground states and unique excitations which can arise instead. A
magnetic system with disorder in bonds often exhibits a short-ranged order, indicating that the system
cannot form a true thermodynamic ground state and thus becomes frustrated. This state of matter, so-
called spin glass, with a multitude of a ground state degeneracy has drawn colossal interest over the
past decades.
Spin glasses are disordered magnetics which characterised by two main characteristics that
strongly distinguish these systems from others: in such systems there is a strong competition between
ferromagnetic and antiferromagnetic interactions, i.e., 'frustrations', and disorder - the freezing (or
solidification) of atoms at different locations during alloy formation. These factors provide key
features of such structures. In such systems with competing interactions, unlike conventional
magnetics, no long-range magnetic order arises with decreasing temperature. But neither does a slow,
gradual freezing of spins occur. Spin glasses have long relaxation times and a rough energy
landscape, so both analytical description and numerical modelling of such systems is challenging. The
processes occurring in such systems cannot be described in terms of classical phase transition theory.
This paper proposes optimized versions of the hybrid algorithm for finding the ground state values
of the Edwards-Anderson model.
VI International Conference Information Technologies and High-Performance Computing (ITHPC-2021),
September 14â16, 2021, Khabarovsk, Russia
EMAIL: kapitan.diu@students.dvfu.ru (Dmitrii Yu. Kapitana)
ORCID: 0000-0001-9815-1891 (Dmitrii Yu. Kapitana)
Šī¸ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
2. Edwards-Anderson model
In 1975 S. Edwards and P. Anderson suggested changing the distribution function of the exchange
interaction in the Ising model to a more complex one, such as the one where the exchange integral đŊđđ
is a random function and the average value of đŊđđ is zero[3].
The interaction đŊđđ between a spin pair (đđ) changes as one goes from one pair to another. The
Hamiltonian is then expressed as:
đģ = ââđŊ đ đ â ââđ , (1)
đđ đ đ đ
âŠđ,đâĒ đ
where đđ , đđ are spins of the system, âŠđ, đâĒ denotes summarizing over the lattice with size N, h -
external magnetic field. The interaction can be ferromagnetic or antiferromagnetic: in the first case,
the interaction arranges spins along one direction; in the second case, the state with antiparallel
direction of spins becomes the most advantageous for the system. The exchange interaction can occur
directly between a pair of magnetic particles (direct exchange interaction), as well as in the presence
of the intermediary particle (indirect exchange interaction). Therefore, the magnitude of the exchange
interaction may strongly depend on the lattice geometry (mutual arrangement of atoms) and the
distance between spins [4].
3. Hybrid Monte-Carlo
Monte Carlo methods, such as the Metropolis or Wang-Landau algorithms, are not only actively
used to study various physical systems [5,6,7,8,9] but also continue to actively develop and improve
due to current Monte Carlo methods have some weaknesses. Single-spin sampling methods suffer
from critical deceleration and applying of multicanonical methods has difficulties in calculating the
thermodynamics of relatively large systems. The use of single-spin Monte-Carlo methods (e.g. the
Metropolis algorithm) to calculate the ground state of systems with coarse energy landscapes is
problematic [10]. To overcome the large energy barriers separating the quasi- degenerated
configurations of the frustrated Ising magnetic, which prevent one from finding their energy-preferred
low-energy states, applying of quasi-Markov processes in the thermodynamics of multi-spin clusters
is required.
To solve the problem of thermodynamics of frustrated vector models of complex systems with
many interacting bodies, searching for ground state configurations, we propose new optimizations for
the Hybrid multi-spin method, described in [11].
3.1. HMC with Monte-Carlo inside kernel
First, authors tried to divide the lattice into sub-lattices with modulation inside such kernels. The
algorithm is worked as follows:
âĸ Spin lattice with periodic boundary conditions is created
âĸ For each spin from the lattice the neighbours are defined
âĸ The initial energy calculation is performed
âĸ Spin lattice is divided on several sublattices, as shown in Figure 1
âĸ Inside those small areas Monte Carlo simulation is started
âĸ The configuration with the lowest energy in kernel is taken
âĸ Kernels are moved in lattice
âĸ After the termination of n cycles, the algorithm is stopped
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Figure 1: Dividing of lattice into sublattices
This algorithm has shown a good efficiency, however accuracy of this method in searching
Ground State is still inappropriate. This is the reason for another suggestion.
3.2. HMC with exact solution inside kernel
Next assumption was to choose spins as midpoint of kernel with the highest energy. The work of
this algorithm is presented below: An example of numbered list is as following.
1. Spin lattice with periodic boundary conditions is created
2. For each spin from the lattice the neighbours are defined
3. The initial energy calculation is performed
4. Spins are randomly chosen from the list of spins with max energy as shown in Figure 2
5. The energy and magnetization of all possible configurations of kernels and the boundary
block of spins are computed by brute force algorithm
Step 5 is repeated until thermodynamic equilibrium is reached in the system. The criterion for
stopping the algorithm can be a given number of iterations or reaching a given temperature value.
Figure 2: Example of choosing a spin with max energy
196
4. Results
To compare the two algorithms, a program was created on the C++. The algorithms were tested on
a square lattice of the two-dimensional Edwards-Anderson model, where bonds have had bimodal
distribution, i.e. amount of ferromagnetic and antiferromagnetic bonds was equal. To calculate the
ground state of different systems, the number of spins was set as N = 6x6, 10x10, 20x20, 30x30.
Calculations were carried out for a supercomputer cluster. To compare the results, the data were
obtained using the algorithm of exact solution, and the parallel tempering algorithm. The results are
showed good potential of Hybrid Monte-Carlo with exact solution in kernel in searching ground states
of frustrated models, see Table 1. Hybrid Monte-Carlo with MC in kernel, despite its efficiency, had
not shown an appropriate result. This can probably be explained by problems with defining direction
of spins on the border between sublattices. After choosing the algorithm, authors started to investigate
the behaviour of staggered magnetization as a function of different values of the external magnetic
field on the example of Edwards-Anderson model with size N = 6x6. The results were compared to
algorithm of exact solution, please, check Table 1 3. Also, authors decided to study the values of the
ground state spin excess. The results were compared to algorithm of exact solution, as well, see Table
1 4.
Figure 3: Staggered magnetization retrieved from optimized HMC and exact solution for the system
with N=6x6
Figure 4: Values of the ground state spin excess retrieved from optimized HMC and exact solution
for the system with N=6x6
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Table 1
Comparison of GS values reached by each method
N Exact Parallel HMC with HMC with
Solution Tempering MC step in kernel exact solution step
in kernel
6x6 -1.50 -1.27 -1.33 -1.50
10x10 -1.40 -1.24 -1.32 -1.40
20x20 - -0.98 -1.06 -1.34
30x30 - -0.76 -0.79 -1.34
5. Conclusion
In this paper, algorithms were considered for finding ground states in the Edward-Anderson
model. Authors looked at Hybrid Monte-Carlo algorithms with different approaches during
modulation of kernels: exact solution and Monte-Carlo. A program was also written to compare two
algorithms within the framework of the conditions we are interested in. After that, using the best
approach key characteristics were calculated. On the basis of the obtained results, it can be concluded
that when choosing an algorithm for searching ground states, one should use a Hybrid Monte-Carlo
algorithm with exact solution inside kernel.
In the future, the approach can be extended to the case of a complex sign-variable exchange long-
range interaction. Also, it is interesting to investigate the ground state of three-dimensional Edwards-
Anderson spin glass.
6. Acknowledgements
This work was financially supported by the state task of the Ministry of Science and Higher
Education of Russia No. 0657-2020-0005.
The studies were carried out using the resources of the Center for Shared Use of Scientific
Equipment "Center for Processing and Storage of Scientific Data of the Far Eastern Branch of the
Russian Academy of Sciences" [12], funded by the Russian Federation represented by the Ministry of
Science and Higher Education of the Russian Federation under project No. 075-15-2021-663.
Additional computing resources were provided by the FEFU supercomputer cluster (cluster.dvfu.ru).
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[3] S. F. Edwards, P. W. Anderson, Theory of spin glasses, J. Phys. F., 1975.
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order parameter and critical exponents of the spin glass in the frame of Edwards-Anderson
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[5] K. Nefedev and V. Kapitan, Concentration phase transition and hysteresis phenomena in co-
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