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
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
Meandering stripes in the frustrated J1-J2 Ising model on the honeycomb lattice
We study the frustrated J1-J2 Ising model on the honeycomb lattice with ferromagnetic nearest-neighbor couplings fixed at J1=1 and strong antiferromagnetic next-nearest-neighbor interactions, i.e., J2≤−1/4. Little is known for this range of J2, whereas for less negative values of J2 the system orders ferromagnetically at low temperatures and appears to remain in the Ising universality class. In previous work it was shown that the model has a largely degenerate ground state, and it was conjectured that there is some kind of phase transition. We introduce a complex-valued nematic order parameter, which can differentiate between the high-temperature paramagnetic phase and the observed partially-disordered stripe phase at lower temperatures. Configurations in this phase consist of stripes of spins parallel with respect to one lattice direction, which collectively meander along the remaining two, producing partially disordered ground states. The sharp peaks in the specific heat observed in earlier work only appear when using periodic boundary conditions and are absent for free boundaries. Additionally, we reveal a striking dependence of the behavior on the aspect ratio of the considered samples. Ultimately, even a careful finite-size scaling analysis for J2=−0.5 and J2=−1 is unable to clearly discern between a crossover without any singularities and some form of continuous transition, including the possibility of an infinite-order transition of the Berezinskii-Kosterlitz-Thouless (BKT) type. For J2=−1/4 we find that the system remains disordered at all temperatures and that it exhibits a finite ground-state entropy per site, for which our simulations provide the accurate asymptotic estimate S(T=0)/N=0.23096093(14) in the thermodynamic limit N→∞.
We study the Ising model with competing ferromagnetic nearest- and antiferromagnetic next-nearest-neighbor interactions of strengths J 1 > 0 and J 2 < 0 , respectively, on the honeycomb lattice. For J 2 > − J 1 / 4 it has a ferromagnetic ground state, and previous work has shown that at least for J 2 ≳ − 0.2 J 1 the transition is in the Ising universality class. For even lower J 2 some indicators pointing towards a first-order transition were reported. By utilizing population annealing Monte Carlo simulations together with a rejection-free and adaptive update, we can equilibrate systems with J 2 as low as − 0.23 J 1 . By means of a finite-size scaling analysis we show that the system undergoes a second-order phase transition within the Ising universality class at least down to J 2 = − 0.23 J 1 and, most likely, for all J 2 > − J 1 / 4 . As we show here, there exist very long-lived metastable states in this system explaining the first-order like behavior seen in only partially equilibrated systems.
To demonstrate the application of computational physics in modern statistical physics, this paper investigates the critical behavior of a finite-size ferromagnetic system based on the two-dimensional square-lattice Ising model.A nearest-neighbor Ising Hamiltonian is constructed, and periodic boundary conditions are adopted to reduce finite-boundary effects.The Metropolis Monte Carlo algorithm is used to sample spin configurations at different temperatures, while energy, magnetization, specific heat, magnetic susceptibility and Binder cumulant are calculated to describe the transition from an ordered state to a disordered state.The simulation results show that the system exhibits clear long-range order at low temperatures, where the order parameter remains high.As temperature increases, thermal fluctuations become stronger, magnetization decreases rapidly, and the system gradually evolves into a disordered state.The specific heat curve shows a pronounced peak at approximately (T = 2.269), while the susceptibility peak shifts slightly toward the high-temperature side due to finite-size effects and the discrete temperature grid.Overall, the simulated critical region is consistent with the theoretical critical temperature of the two-dimensional square-lattice Ising model.Spin-configuration analysis further reveals enhanced fluctuations and domain reconstruction near the critical region.In addition, a thermalization criterion and an extended sampling test for L = 48 near the critical region are added to examine the influence of finite sampling and equilibration.These results indicate that Monte Carlo simulation of the two-dimensional Ising model can effectively reproduce continuous phase transition, critical fluctuation and finite-size effects, providing a basic computational case for understanding critical phenomena in complex systems and data-driven physical analysis.
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.
Classical and quantum spin systems are widely used models in both experimental and theoretical condensed matter physics. In many materials, the electronic interactions can be difficult to model exactly. However, in some insulators (Mott insulators), the magnetic (spin) interactions can be captured well with spin-only models. Several models are studied in this thesis. First, I report the solution of a long-standing issue in a classical frustrated spin model (i.e., where the quantum effects are neglected and the complexity is due to competing interactions): the nature of the thermal phase transition to a stripe state in the two dimensional (2D) J1-J2 model. Here J1 and J2 are nearest- and next-nearest-neighbor couplings. Monte Carlo simulations with single-spin updates are used for the calculations, and an extended-ensemble method, a generalization of the Wang-Landau algorithm, is also developed and tested. I focus on the study of "weak universality" behavior (continuously varying critical exponents, with one of the exponents staying fixed), for which I show a correspondence with a known class of conformal field theories with charge c = 1. Next, moving to quantum spins, to shed light on magnetic systems studied experimentally and to investigate new types of quantum states of interest in developing theories of quantum magnetism, I study the S = 1/2 Heisenberg model on the 2D square lattice with added six-spin interactions (the so-called J-Q3 model) as well as a set of 3D quantum antiferromagnets on dimerized lattices. Here I use the stochastic series expansion quantum Monte Carlo method. In the study of the J-Q3 model, I report on a similar weak-universality behavior as in the classical J1-J2 model, but with a mapping to a different known class of c = 1 conformal field theories. The critical behavior of the system again shows continuously changing exponents, in a way which corresponds to a gradual weakening of Z4 symmetry-breaking to an emergent U(1) symmetry. In the study of dimerized antiferromagnets, I report on a universal behavior of the Néel temperature TN, which can be related to ground state parameters independently of the microscopic interaction details in several different models.
We consider the thermal phase transition from a paramagnetic to a stripe-antiferromagnetic phase in the frustrated two-dimensional square-lattice Ising model with competing interactions J1<0 (nearest neighbor, ferromagnetic) and J2>0 (second neighbor, antiferromagnetic). The striped phase breaks a Z4 symmetry and is stabilized at low temperatures for g=J2/∣J1∣>1/2. Despite the simplicity of the model, it has proved difficult to precisely determine the order and the universality class of the phase transitions. This was done convincingly only recently by Jin et al. [Phys. Rev. Lett. 108, 045702 (2012)]. Here, we further elucidate the nature of these transitions and their anomalies by employing a combination of cluster mean-field theory, Monte Carlo simulations, and transfer-matrix calculations. The J1-J2 model has a line of very weak first-order phase transitions in the whole region 1/2<g<g∗, where ${g}^{*}=0.67\ifmmode\pm\else\textpm\fi{}0.01$. Thereafter, the transitions from g=g∗ to $g\ensuremath{\rightarrow}\ensuremath{\infty}$ are continuous and can be fully mapped, using universality arguments, to the critical line of the well-known Ashkin-Teller model from its four-state Potts point to the decoupled Ising limit. We also comment on the pseudo-first-order behavior at the Potts point and its neighborhood in the Ashkin-Teller model on finite lattices, which in turn leads to the appearance of similar effects in the vicinity of the multicritical point g∗ in the J1-J2 model. The continuous transitions near g∗ can therefore be mistaken for first-order transitions, and this realization was the key to understanding the paramagnetic-striped transition for the full range of g>1/2. Most of our conclusions are based on Monte Carlo calculations, while the cluster mean-field and transfer-matrix results provide useful methodological benchmarks for weakly first-order behaviors and Ashkin-Teller criticality.