Tensor network investigation of frustrated Ising models

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.

Submonolayer growth of cobalt on metallic and insulating surfaces studied by scanning tunneling microscopy and kinetic Monte-Carlo simulations

Philipp Buluschek

Submonolayer epitaxial growth is obtained by the deposition of less than a complete layer of atoms on a single crystal surface. It is of fundamental interest for industrial applications (e.g. in the semiconductor industry) as well as from the point of view of basic research. For example, it is known that nanometer-sized atomic structures (nanostructures) exhibit remarkable physical and chemical properties which can differ greatly from those of bulk matter. In order to investigate these properties, it is often necessary to create large quantities of well defined and possibly spatially ordered nanostructures. One way to achieve this result is self-organized growth where one deposits atoms on a clean crystalline surface and lets the growth process evolve freely. Here, nanostructures result from the atomic diffusion and aggregation processes taking place at the surface. Understanding the exact nature of these processes is of ongoing interest in the field of nanostructure growth. In this thesis we report about submonolayer growth experiments of the ferromagnetic transition metal cobalt. Cobalt was chosen because it exhibits remarkable magnetic properties. These experiments were performed in an ultra-high vacuum chamber and molecular beam epitaxy was used to grow the nanostructures. Observations of the surface were made using a variable-temperature scanning tunneling microscope (STM). In order to get a better understanding of the atomic processes happening at the surface, we developed two adapted computational simulation methods. Kinetic Monte-Carlo (KMC) simulations were used to get an atomistic picture of the surface while mean field rate equations were integrated numerically to yield cluster densities. We study the growth of cobalt on three different surfaces. By depositing Co on a clean Pt(111) crystal surface, we observe that the platinum surface reconstructs by forming star-shaped partial dislocations for sample temperatures above 180 K (-93°C). We also observe that island densities deviate from predictions of all known models towards higher values for these same temperatures. By simulation we are able to show that insertion of Co into the topmost platinum layer creates a repulsive network of dislocations. We show that these dislocations act as diffusion inhibiting barriers and thus influence the island density by constraining the free movement of atoms at the surface. We also show that the Co dimer and trimer diffuse on Pt(111) before dissociating and are able to extract corresponding activation barriers. We also study the deposition of Co on a Ru(0001) surface which shows a more conventional temperature dependence. By comparing simulation and experiment we extract all relevant diffusion activation barriers and show that at high temperatures the Co dimer and trimer dissociate. Finally, we investigate the growth of Co on the hexagonal boron nitride superlattice which forms on a Rh(111) surface. On this surface Co clusters form three-dimensional hemispherical dots. We show how the substrate corrugation influences diffusion of Co atoms and that clusters up to the size of the pentamer can diffuse. By simulation, we also extract the relevant migrations and desorption barriers.

EPFL2008

Uncovering the Physics of Frustrated Quantum Magnets using the Correlation Density Matrix Approach

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.

EPFL2010

Effects of magnetic field on the cuprate high-T-c superconductor La2-xSrxCuO4

Henrik Moodysson Rønnow

This article discusses neutron scattering measurements on the cuprate, high transition temperature superconductor La2-xSrxCuO4 (LSCO) in an applied magnetic field. LSCO is a type-II superconductor and magnetic flux can penetrate the material via the formation of vorticies. Phase coherent superconductivity characterized by zero resistance is suppressed to the lower field-dependent irreversibility temperature (T-irr(H)) and occurs when the vortices freeze into a lattice. Because superconductivity is destroyed within the vortex cores, an investigation of the vortex state provides information about the ground state that would have appeared had superconductivity not intervened. Our measurements reveal that both optimally doped LSCO (x = 0.16, T-c = 38.5 K) and underdoped LSCO (x = 0.10, T-c = 29 K) have an enhanced antiferromagnetic response in a field. Measurements of the optimally doped system for H = 7.5 T show that inelastic sub-gap spin fluctuations first disappear with the loss of finite resistivity at T-irr, but then reappear at a lower temperature with increased lifetime and correlation length compared to the normal state. In the underdoped system elastic antiferromagnetism develops below T-c in zero field, and is significantly enhanced by application of a magnetic field; phase coherent superconductivity is then established within the anti- ferromagnetic phase at T-irr.

2004

Submonolayer growth of cobalt on metallic and insulating surfaces studied by scanning tunneling microscopy and kinetic Monte-Carlo simulations

Philipp Buluschek

EPFL2008

Uncovering the Physics of Frustrated Quantum Magnets using the Correlation Density Matrix Approach

EPFL2010