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Tensor network investigation of frustrated Ising models

Jeanne Colbois

Published Jan 1, 2022• 0 citations
Publisher

Abstract

In spin systems, geometrical frustration describes the impossibility of minimizing simultaneously all the interactions in a Hamiltonian, often giving rise to macroscopic ground-state degeneracies and emergent low-temperature physics. In this thesis, combining tensor network (TN) methods to Monte Carlo (MC) methods and ground-state energy lower bound approaches, we study two-dimensional frustrated classical Ising models. In particular, we focus on the determination of the residual entropy in the presence of farther-neighbor interactions in kagome lattice Ising antiferromagnets (KIAFM). In general, using MC to determine the residual entropy is a significant challenge requiring ad-hoc updates, a precise evaluation of the energy at all temperatures to allow for thermodynamic integration, and a good control of the finite-size scaling behavior. As an alternative, we turn to TNs; however, we argue that, in the presence of frustration and macroscopic ground-state degeneracy, standard algorithms fail to converge at low temperatures on the usual TN formulation of partition functions. Inspired by methods for constructing ground-state energy lower bounds, we propose a systematic way to find the ground-state local rule using linear programming. Characterizing the rules as tiles that can be tessellated to form ground states of the model gives rise to a natural contractible TN formulation of the partition function. This method provides a direct access to the ground-state properties of frustrated models and, in particular, allows an extremely precise determination of their residual entropy. We then study two models inspired by artificial spin systems on the kagome lattice with out-of-plane (OOP) anisotropy. The first model is motivated by experiments on an array of chirally coupled nanomagnets. We argue that the farther-neighbor to nearest-neighbor couplings ratios in this system are much smaller than in the dipolar case, J2/J1 being of the order of 2%. A comparison of the experimental correlations with the results of extensive TN and MC simulations shows that (1) the experimental second- and third-neighbor correlations are inverted as compared to those of a pure nearest-neighbor model at equilibrium (even with a magnetic field), and (2) second-neighbor couplings as small as 1% of the nearest-neighbor couplings will affect the spin-spin correlations even at fairly high temperatures. Motivated by dipolar coupled artificial spin systems, we turn to the progressive lifting of the ground-state degeneracy of the KIAFM. We provide a detailed study of the ground-state phases of this model with up to third neighbor interactions, for arbitrary J2, J3 such that J1 >> J2, J3, obtaining exact results for the ground-state energies. When all couplings are antiferromagnetic, we exhibit three macroscopically degenerate ground-state phases and establish their residual entropy using our TN approach. Furthermore, in the phase corresponding to the dipolar KIAFM truncated to third neighbors, we use the ground-state tiles to establish the existence of a mapping to the ground-state manifold of the triangular Ising antiferromagnet.

DOI
10.5075/epfl-thesis-9290

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In spin systems, geometrical frustration describes the impossibility of minimizing simultaneously all the interactions in a Hamiltonian, often giving rise to macroscopic ground-state degeneracies and emergent low-temperature physics. In this thesis, combining tensor network (TN) methods to Monte Carlo (MC) methods and ground-state energy lower bound approaches, we study two-dimensional frustrated classical Ising models. In particular, we focus on the determination of the residual entropy in the presence of farther-neighbor interactions in kagome lattice Ising antiferromagnets (KIAFM). In general, using MC to determine the residual entropy is a significant challenge requiring ad-hoc updates, a precise evaluation of the energy at all temperatures to allow for thermodynamic integration, and a good control of the finite-size scaling behavior. As an alternative, we turn to TNs; however, we argue that, in the presence of frustration and macroscopic ground-state degeneracy, standard algorithms fail to converge at low temperatures on the usual TN formulation of partition functions. Inspired by methods for constructing ground-state energy lower bounds, we propose a systematic way to find the ground-state local rule using linear programming. Characterizing the rules as tiles that can be tessellated to form ground states of the model gives rise to a natural contractible TN formulation of the partition function. This method provides a direct access to the ground-state properties of frustrated models and, in particular, allows an extremely precise determination of their residual entropy. We then study two models inspired by artificial spin systems on the kagome lattice with out-of-plane (OOP) anisotropy. The first model is motivated by experiments on an array of chirally coupled nanomagnets. We argue that the farther-neighbor to nearest-neighbor couplings ratios in this system are much smaller than in the dipolar case, J2/J1 being of the order of 2%. A comparison of the experimental correlations with the results of extensive TN and MC simulations shows that (1) the experimental second- and third-neighbor correlations are inverted as compared to those of a pure nearest-neighbor model at equilibrium (even with a magnetic field), and (2) second-neighbor couplings as small as 1% of the nearest-neighbor couplings will affect the spin-spin correlations even at fairly high temperatures. Motivated by dipolar coupled artificial spin systems, we turn to the progressive lifting of the ground-state degeneracy of the KIAFM. We provide a detailed study of the ground-state phases of this model with up to third neighbor interactions, for arbitrary J2, J3 such that J1 >> J2, J3, obtaining exact results for the ground-state energies. When all couplings are antiferromagnetic, we exhibit three macroscopically degenerate ground-state phases and establish their residual entropy using our TN approach. Furthermore, in the phase corresponding to the dipolar KIAFM truncated to third neighbors, we use the ground-state tiles to establish the existence of a mapping to the ground-state manifold of the triangular Ising antiferromagnet.
Motivated by the recent success of tensor networks to calculate the residual entropy of spin ice and kagome Ising models, we develop a general framework to study frustrated Ising models in terms of infinite tensor networks that can be contracted using standard algorithms for infinite systems.This is achieved by reformulating the problem as local rules for configurations on overlapping clusters chosen in such a way that they relieve the frustration, i.e., that the energy can be minimized independently on each cluster.We show that optimizing the choice of clusters, including the weight on shared bonds, is crucial for the contractibility of the tensor networks, and we derive some basic rules and a linear program to implement them.We illustrate the power of the method by computing the residual entropy of a frustrated Ising spin system on a kagome lattice with next-next-nearest-neighbor interactions, vastly outperforming Monte Carlo methods in speed and accuracy.The extension to finite temperatures is briefly discussed.
Despite their simple formulation, short-range classical antiferromagnetic Ising models on frustrated lattices give rise to exotic phases of matter, in particular, due to their macroscopic ground-state degeneracy. Recent experiments on artificial spin systems comprising arrays of chirally coupled nanomagnets provide a significant strengthening of the nearest-neighbor couplings compared to systems with dipolar-coupled nanomagnets. This opens the way to design artificial spin systems emulating Ising models with nearest-neighbor couplings. In this paper, we compare the results of an extensive investigation with tensor network and Monte Carlo simulations of the nearest- and farther-neighbor $({J}_{1}\ensuremath{-}{J}_{2}\ensuremath{-}{J}_{3||})$ kagome Ising antiferromagnet with the experimental spin-spin correlations of a kagome lattice of chirally coupled nanomagnets. Even though the ratios between the farther-neighbor couplings and the nearest-neighbor coupling estimated from micromagnetic simulations are much smaller than for dipolar-coupled nanomagnets, we show that they still play an essential role in the selection of the correlations.
Frustration is a ubiquitous phenomenon in many-body physics that influences the nature of the system in a profound way with exotic emergent behavior. Despite its long research history, the analytical or numerical investigations on frustrated spin models remain a formidable challenge due to their extensive ground-state degeneracy. In this paper, we propose a unified tensor network theory to numerically solve the frustrated classical spin models on various two-dimensional (2D) lattice geometry with high efficiency. We show that the appropriate encoding of emergent degrees of freedom in each local tensor is of crucial importance in the construction of the infinite tensor network representation of the partition function. The frustrations are thus relieved through the effective interactions between emergent local degrees of freedom. Then the partition function is written as a product of a one-dimensional (1D) transfer operator, whose eigenequation can be solved by the standard algorithm of matrix product states rigorously, and various phase transitions can be accurately determined from the singularities of the entanglement entropy of the 1D quantum correspondence. We demonstrated the power of our general theory by numerically solving 2D fully frustrated XY spin models on the kagome, square, and triangular lattices, giving rise to a variety of thermal phase transitions from infinite-order Brezinskii-Kosterlitz-Thouless transitions, second-order transitions, to first-order phase transitions. Our approach holds the potential application to other types of frustrated classical systems like Heisenberg spin antiferromagnets.
We develop a technique for calculating three-dimensional classical partition functions using projected entangled-pair states (PEPS). Our method is based on variational PEPS optimization algorithms for two-dimensional quantum spin systems, and allows us to compute free energies directly in the thermodynamic limit. The main focus of this work is classical frustration in three-dimensional many-body systems leading to an extensive ground-state degeneracy. We provide high-accuracy results for the residual entropy of the dimer model on the cubic lattice, water ice Ih{I}_{h}Ih​, and water ice Ic{I}_{c}Ic​. In addition, we show that these systems are in a Coulomb phase by computing the dipolar form of the correlation functions. As a further benchmark of our methods, we calculate the critical temperature and exponents of the Ising model on the cubic lattice.
Frustrated spin systems exhibit a wide range of phenomena such as novel phases, residual entropy and glassy dynamics. In contrast to the usual assumption of universality, microscopic details such as the underlying lattice structure often play a crucial role in these systems. In this dissertation the frustrated J1-J2 Ising model on the honeycomb lattice is considered. Earlier work shows that the coupling ratio R = J2 /|J1 | = −1/4 separates regions of two different ground states. Above this threshold the ground state is (anti-)ferromagnetically ordered for (anti-)ferromagnetic J1 , whereas below it infinitely many states of minimal energy exist. A recent Monte Carlo (MC) study by Žukovič finds a continuous transition with Ising critical exponents for R ≥ −0.2, leaving open the question of criticality for −1/4 < R < −0.2, for which the study was obfuscated by the presence of strong metastability. The regime with the highly degenerate ground states (R < −1/4), on the other hand, is characterized by feature-rich specific heat curves, and slow low-temperature dynamics reminiscent of spin glasses, although there is no disorder in this system. Prior to this work, no order parameter has been identified, and there is no clear understanding of its potentially critical behavior. Here, the question of criticality for −1/4 < R < −0.2 is addressed, and Ising universality is found for lower values of R, namely, at least for R ≥ −0.23. This result is achieved by carrying out finite-size scaling analyses of response functions like the magnetic susceptibility, and of the complex partition function zeros. In both cases, the underlying numerical data was generated using population annealing (PA) MC, a framework devised for spin glasses but generally applicable to systems with metastability. Furthermore, the region of degenerate ground states is studied using different boundary conditions, which are found to strongly affect the thermodynamic observations. A broken symmetry at zero temperature is identified, which can be quantified using a complex nematic order parameter. The MC data are consistent with a continuous symmetry-breaking transition. However, due to the limited system sizes that could be studied, and the presence of strong finite-size effects, one cannot rule out a higher-order transition or a cross-over without any singularities. In the process of studying this system, also the understanding of the PA framework was further developed. Specifically, the role of the resampling step in the algorithm was investigated. While it is crucial for efficient equilibration, no benefit is found in noisier resampling schemes such as the multinomial resampling method. Instead, two resampling schemes, which have the lowest sampling variance, are identified as preferential.:1 Introduction I Background and Methodology 2 Background 2.1 Fundamentals of Statistical Mechanics 2.1.1 Equilibrium ensembles 2.1.2 Phase transitions and critical behavior 2.1.3 Common spin models 2.2 Frustrated Spin Models 2.2.1 Frustrated J1-J2 Ising model on the square lattice 2.2.2 Gonihedric model in two dimensions 2.3 Monte Carlo Simulations 2.3.1 Fundamentals 2.3.2 Autocorrelation and error bars in MCMC time series 2.3.3 Possible implementations 2.3.4 Generalized ensemble methods 3 Methodology 3.1 General Algorithmic Design 3.2 Population Annealing 3.2.1 Algorithm 3.2.2 Assessing population annealing performance 3.2.3 Adaptive temperature steps 3.2.4 Weighted averages 3.2.5 Population annealing and first-order transitions 3.3 Estimating Critical Exponents from Finite Systems 3.3.1 Standard finite size scaling 3.3.2 Complex partition function zeros II Results 4 Algorithmic Improvements and Extensions to the Population Annealing Framework 4.1 Resampling Step in Population Annealing 4.1.1 Conceptual overview 4.1.2 Possible resampling methods 4.1.3 Benchmarking different resampling methods 4.1.4 Simulation results 4.1.5 PA simulations with finite populations in the limit of perfect equilibration 4.1.6 PA simulations in the limit of infinite populations and perfect equilibration 4.1.7 Conclusion 4.2 Using Population Annealing on Many Computing Nodes 4.2.1 Literature overview and different rebalancing schemes 4.2.2 Simulations of simulations 4.2.3 Conclusion and outlook 4.3 Adaptive Number of Spin Updates in Population Annealing 4.4 Microcanonical Population Annealing 5 The Frustrated J1-J2 Ising Model on the Honeycomb Lattice 5.1 Overview 5.2 Regime of Ising Universality Class 5.2.1 Finite-size scaling analysis 5.2.2 Energy barriers 5.2.3 Exploring the complex partition function zeros 5.3 Finite Ground State Entropy per Spin for J2/|J1| = −1/4 5.4 Highly Degenerate Striped Phase for J2/|J1| < −1/4 5.4.1 Boundary conditions 5.4.2 Energy minimization and ground states 5.4.3 Topological defects 5.4.4 Observables 5.4.5 Simulation details 5.4.6 Simulation results for R = −0.5 5.4.7 Simulation results for R = −1 5.4.8 Discussion 5.4.9 Conclusion and outlook 6 Parallel tempering and its infinite-swap limit 6.1 Formalism 6.2 Details on Implementation 6.3 Numerical Results from INS-PT Simulations 6.3.1 One-dimensional Ising model 6.3.2 Two-dimensional Ising model 6.3.3 q-state Potts model 6.3.4 Lennard-Jones clusters 6.4 Discussion 7 Conclusions and outlook Bibliography Appendices A Algorithmic Improvements and Extensions to the Population Annealing framework: Further details A.1 Generating correlated energy time series B J1-J2 honeycomb Ising model: Further plots and details B.1 GPU domain decomposition B.2 Finite-size scaling using Metropolis population annealing B.3 The decoupled antiferromagnet limit B.4 Zero temperature value of the nematic order parameter B.5 Percolation properties for J2 = −0.5 B.6 Binder parameter of the one-dimensional Ising model C Application for Ethical Approval
In this thesis, we investigate frustrated systems with focus on the triangular lattice anisotropic Heisenberg antiferromagnet. Firstly, we examined the material CeMgAl11O19 as a potential quantum spin liquid utilizing the density matrix renormalization group method (DMRG). Two different low-temperature effective models have been proposed. The first was motivated by similar systems and contains long-range exchange interac- tion. The second, introduced much later, assumes only nearest-neighbor interactions and exhibits strong anisotropy, mixing ferromagnetic and antiferromagnetic couplings. We showed that the first model does not reproduce the experimental results even qual- itatively, but the second is consistent with the measured data. Our analysis demon- strated that CeMgAl11O19 is farther away from the critical point than previously claimed in the literature. We then investigated the second model in more detail and examined its complicated ground state phase diagram in magnetic field. The complicated struc- ture of the diagram was analyzed by unsupervised machine learning techniques as well as by employing order parameters constructed from structure factors. Finally, we used the recently introduced variational Monte Carlo based on neural network quantum states (NNQS) to investigate the ground state of...