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Publication# Unconventional quantum ordered and disordered states in the highly frustrated spin-1/2 Ising-Heisenberg model on triangles-in-triangles lattices

Abstract

The spin-1/2 Ising-Heisenberg model on two geometrically related triangles-in-triangles lattices is exactly solved through the generalized star-triangle transformation, which establishes a rigorous mapping correspondence with the effective spin-1/2 Ising model on a triangular lattice. Basic thermodynamic quantities were exactly calculated within this rigorous mapping method along with the ground-state and finite-temperature phase diagrams. Apart from the classical ferromagnetic phase, both investigated models exhibit several unconventional quantum ordered and disordered ground states. A mutual competition between two ferromagnetic interactions of basically different character generically leads to the emergence of a quantum ferromagnetic phase, in which a symmetric quantum superposition of three up-up-down states of the Heisenberg trimers accompanies a perfect alignment of all Ising spins. In the highly frustrated regime, we have either found the disordered quantum paramagnetic phase with a substantial residual entropy or a similar but spontaneously ordered phase with a nontrivial criticality at finite temperatures. The latter quantum ground state exhibits a striking coexistence of imperfect spontaneous order with partial disorder, which is evidenced by a quantum reduction of the spontaneous magnetization of Heisenberg spins that indirectly causes a small reduction of the spontaneous magnetization of otherwise classical Ising spins. DOI: 10.1103/PhysRevB.87.024421

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This work is devoted to the study, the development, and the application of a new systematic method yielding the dominant correlations that govern a quantum many-body state in an unbiased way. The dominant correlations between any two disjoint blocks of a system are extracted by performing a singular value decomposition of the correlation density matrix (CDM) between those blocks. We determine several mathematical properties and features of this method, in particular the consequences of the lattice symmetries or the symmetries intrinsic to the studied state on the singular values spectrum. We investigate the relation between the norm of the CDM – providing a natural measure of the total correlation between the two blocks – and the so-called mutual information, a quantity originally introduced in quantum information theory. This novel tool is utilized for sheding new light on the zero temperature physics of the spin-1/2 frustrated ferromagnetic J1–J2 Heisenberg chain in a magnetic field as well as on the low-energy physics of the spin-1/2 antiferromagnetic Heisenberg model on the two-dimensional kagomé lattice. The states are computed using exact diagonalization and the density matrix renormalization group procedure in the first case, and exact diagonalization only in the second case. This work is introduced in Chapter 1. The first model is then presented in Chapter 2. Chapter 3 introduces the CDM method, and Chapter 4 is devoted to the study of the kagomé antiferromagnet. In the J1–J2 chain, we reveal a vector chiral phase at low magnetic field and a sequence of multipolar Luttinger liquid phases at high field. We explicitly show that these multipolar phases result from the destabilization – driven by a locking mechanism – of the classical spiral ground state in the absence of magnetic field. This point of view is completely new: multipolar phases were known to be a possible destabilization of ferromagnetic phases, but they have never been reported as a destabilization of spiral states yet. Regarding the kagomé antiferromagnet, we address for the first time the question of the nature of the singlet states forming its quite dense singlet spectrum above the ground state. We show that some of these low-lying singlet states have large dimer correlations which do not seem to significantly decrease with the distance, moreover our CDM studies confirm that the dominant correlations in those singlet states are of the dimer-dimer type. Studies of Von Neumann block entropies reveal a very short correlation length on the one hand, and entropies that are roughly independent on the energy of the state under consideration on the other hand. The scenario of a valence bond crystal phase is investigated and the relevance of different kinds of crystals (from the literature or ad hoc) for reproducing the dimer correlations in the 36-site sample is probed.

Magnetism is largely responsible for the body centered cubic to face centered cubic structural phase transition occurring in iron at 1185 K and to many anomalies in the vicinity of the ferromagnetic to paramagnetic phase transition at 1043 K, as for instance an anomalous softening of the tetrahedral shear modulus. Current atomistic models including magnetism are either limited to the treatment of perfect lattice models or to zero temperatures, while research and development of candidate materials for future fission and fusion power plants requires the modeling of irradiation induced defects in ferritic/martensitic steels at high temperatures. An attempt to fill this gap is the Dudarev-Derlet potential, which includes zero temperature magnetism in an embedded atom method formalism, together with a more recent extension of the method to the inclusion of spin rotations at non zero temperature with nearly half the computational speed of an embedded atom method potential. In this work, we report on the optimization of the Dudarev-Derlet potential to the zero temperature bulk properties of the non-magnetic and ferromagnetic bcc and fcc phases, including the third order elastic constants of the ferromagnetic bcc phase, the point defects formation and migration energies and the core structure of the screw dislocation with Burgers vector 1/2[111], either from experiments or from density functional theory calculations, where we develop a method to fit the core structure of the screw dislocation based on the Suzuki-Takeuchi model. Three representative fits from the optimization of the Dudarev-Derlet potential are compared with recent semi empirical potentials for iron, with density functional theory and experiments. The migration energies of the self-interstitial range from 0.31 eV to 0.42 eV, compared to a density functional theory value close to 0.35 eV and an experimental value close to 0.3 eV, and the vacancy migration energies range from 0.85 eV to 0.94 eV, compared to a density functional theory value close to 0.65 eV. Clusters composed of parallel self-interstitials are oriented along ‹110› if the number of interstitials composing the cluster is smaller or equal than 3, while for bigger clusters the ‹111› orientation is more stable, in qualitative agreement with density functional theory. Depending on which one of the three representative fits is chosen, the formation entropy of one ‹110› dumbbell calculated by the thermodynamical integration method in the range from 300 K to 600 K varies from 0.28 kB to 4.02 kB. The diffusion coefficient of the ‹110› dumbbell at 600 K ranges from 1×10-6 cm2/s to 10×10-6 cm2/s, while at room temperatures the scatters extends over three orders of magnitude. The main difficulties, common to all the semi empirical potentials considered in the work, are related to the description of the fcc phase and the migration mechanism of the screw dislocation. The semi empirical potentials are unable to distinguish the anti-ferromagnetic fcc from the low spin ferromagnetic fcc or the high spin ferromagnetic fcc. Considering the equilibrium volume and the bulk magnetic moment, the high spin phase is the one which most resembles the ferromagnetic fcc phase of the Dudarev-Derlet potentials. Finally, for those fits with a non-degenerate core structure, we investigate some fundamental aspects of the migration mechanism of the screw dislocation with Burgers vector 1/2[111] at zero temperature and at zero applied stress, by calculating the Peierls potential in the [211] direction between two structurally equivalent soft cores. This confirms the existence of a stable core structure in the middle of the migration path not observed in density functional theory, which is actually found to be energetically degenerate with the soft core. The consequences of this are discussed in terms of formation energies of double kinks in the [211] direction.

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.