Residual entropy | EPFL Graph Search

Residual entropy is the difference in entropy between a non-equilibrium state and crystal state of a substance close to absolute zero. This term is used in condensed matter physics to describe the entropy at zero kelvin of a glass or plastic crystal referred to the crystal state, whose entropy is zero according to the third law of thermodynamics. It occurs if a material can exist in many different states when cooled. The most common non-equilibrium state is vitreous state, glass. A common example is the case of carbon monoxide, which has a very small dipole moment. As the carbon monoxide crystal is cooled to absolute zero, few of the carbon monoxide molecules have enough time to align themselves into a perfect crystal, (with all of the carbon monoxide molecules oriented in the same direction). Because of this, the crystal is locked into a state with 2^N different corresponding microstates, giving a residual entropy of S=Nk\ln(2), rather than zero. Another

Tensor network investigation of frustrated Ising models

Jeanne Colbois

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.

EPFL2022

Solving frustrated Ising models using tensor networks

Jeanne Colbois, Frédéric Mila

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.

AMER PHYSICAL SOC2021

Low-temperature spin dynamics of a valence bond glass in Ba2YMoO6

Julian Piatek, Henrik Moodysson Rønnow

We carried out ac magnetic susceptibility measurements and muon spin relaxation spectroscopy on the cubic double perovskite Ba2YMoO6, down to 50 mK. Below similar to 1 K the muon relaxation is typical of a magnetic insulator with a spin-liquid type ground state, i.e. without broken symmetries or frozen moments. However, the ac susceptibility revealed a dilute-spin-glass-like transition below similar to 1 K. Antiferromagnetically coupled Mo5+ 4d(1) electrons in triply degenerate t(2g) orbitals are in this material arranged in a geometrically frustrated fcc lattice. Bulk magnetic susceptibility data has previously been interpreted in terms of a freezing to a heterogeneous state with non-magnetic sites where 4d(1) electrons have paired in spin-singlets dimers, and residual unpaired Mo5+ 4d(1) electron spins. Based on the magnetic heat capacity data it has been suggested that this heterogeneity is the result of kinetic constraints intrinsic to the physics of the pure system (possibly due to topological overprotection) leading to a self-induced glass of valence bonds between neighbouring 4d(1) electrons. The muon spin relaxation (mu SR) unambiguously points to a heterogeneous state with a static arrangement of unpaired electrons in a background of (valence bond) dimers between the majority of Mo5+ 4d electrons. The ac susceptibility data indicate that the residual magnetic moments freeze into a dilute-spin-glass-like state. This is in apparent contradiction with the muon-spin decoupling at 50 mK in fields up to 200 mT, which indicates that, remarkably, the time scale of the field fluctuations from the residual moments is similar to 5 ns. Comparable behaviour has been observed in other geometrically frustrated magnets with spin-liquid-like behaviour and the implications of our observations on Ba2YMoO6 are discussed in this context.