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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.

In the field of quantum magnetism, the advent of numerous spin-orbit assisted Mott insulating compounds, such as the family of Kitaev materials, has led to a growing interest in studying general spin models with non-diagonal interactions that do not retain the SU(2) invariance of the underlying spin degrees of freedom. However, the exchange frustration arising from these non-diagonal and often bond-directional interactions for two- and three-dimensional lattice geometries poses a serious challenge for numerical many-body simulation techniques. In this paper, we present an extended formulation of the pseudo-fermion functional renormalization group that is capable of capturing the physics of frustrated quantum magnets with generic (diagonal and off-diagonal) two-spin interaction terms. Based on a careful symmetry analysis of the underlying flow equations, we reveal that the computational complexity grows only moderately, as compared to models with only diagonal interaction terms. We apply the formalism to a kagome antiferromagnet which is augmented by general in-plane and out-of-plane Dzyaloshinskii-Moriya (DM) interactions, as argued to be present in the spin liquid candidate material herbertsmithite. We calculate the complete ground state phase diagram in the strength of in-plane and out-of-plane DM couplings, and discuss the extended stability of the spin liquid of the unperturbed kagome antiferromagnet in the presence of these couplings., Comment: 19 pages, 6 figures

The high-field regime of the spin-s XXZ antiferromagnet on the kagome lattice gives rise to macroscopically degenerate ground states thanks to a completely flat lowest single-magnon band. The corresponding excitations can be localized on loops in real space and have been coined "localized magnons". Thus, the description of the many-body ground states amounts to characterizing the allowed classical loop configurations and eliminating the quantum mechanical linear relations between them. Here, we investigate this loop-gas description on finite kagome lattices with open boundary conditions and compare the results with exact diagonalization for the spin-1/2 XY model on the same lattice. We find that the loop gas provides an exact account of the degenerate ground-state manifold while a hard-hexagon description misses contributions from nested loop configurations. The densest packing of the loops corresponds to a magnon crystal that according to the zero-temperature magnetization curve is a stable ground state of the spin-1/2 XY model in a window of magnetic fields of about 4% of the saturation field just below this saturation field. We also present numerical results for the specific heat obtained by the related methods of thermal pure quantum (TPQ) states and the finite-temperature Lanczos method (FTLM). For a field in the stability range of the magnon crystal, one finds a low-temperature maximum of the specific heat that corresponds to a finite-temperature phase transition into the magnon crystal at low temperatures., Comment: published version; 17 pages of text including 10 figures and Ukrainian abstract

The collaboration between the University of St Andrews and Moscow State University was funded by a Royal Society International Exchanges grant, in collaboration with the Russian Foundation for Basic Research (12-03-92604). Lewis Downie thanks the EPSRC for a Ph.D. studentship via a Doctoral Training grant (EP/P505097/1). This work was carried out with the support of the Diamond Light Source, beamtime application EE7980. Alexandre Vasiliev acknowledges support of RFBR through grants Numbers 13-02-00174, 14-02-92002 and 14-02-92693. Two new members of the A2B′Cu3F12 family of kagome-related materials have been prepared, in order to further understand the crystal-chemical relationships, phase transitions and magnetic behaviour within this family of potentially frustrated S = ½ two-dimensional quantum magnets. Cs2TiCu3F12 adopts a crystal structure with the ideal kagome lattice topology (space group R m) at ambient temperature. Diffraction studies reveal different symmetry-lowering structural phase transitions in single crystal and polycrystalline forms at sub-ambient temperatures, with the single crystal form retaining rhombohedral symmetry and the powder form being monoclinic. In both cases, long-range antiferromagnetic order occurs in the region 16–20 K. Rb2TiCu3F12 adopts a distorted triclinic structure even at ambient temperatures. Publisher PDF

We provide strong numerical evidence, using improved variational wave functions, for a ground state with vanishing spin gap in the spin-$1/2$ quantum Heisenberg model on the kagome lattice. Starting from the algebraic $U(1)$ Dirac spin liquid state proposed by Y. Ran $et al.$ [Phys. Rev. Lett. $98$, $117205$ ($2007$)] and iteratively applying a few Lanczos steps, we compute the lowest $S=2$ excitation constructed by exciting spinons close to the Dirac nodes. Our results are compatible with a vanishing spin gap in the thermodynamic limit and in consonance with a power-law decay of long distance spin-spin correlations in real space. The competition with a gapped (topological) spin liquid is discussed., Comment: 5 pages, 3 figures, 2 tables. Published version

The hole-doped kagome lattice of Cu${}^{2+}$ ions in LaCuO${}_{2.66}$ was investigated by nuclear quadrupole resonance, electron spin resonance, electrical resistivity, bulk magnetization, and specific-heat measurements. For temperatures above $\ensuremath{\sim}180$ K, the spin and charge properties show an activated behavior suggestive of a narrow-gap semiconductor. At lower temperatures, the results indicate an insulating ground state which may or may not be charge ordered. While the frustrated spins in remaining patches of the original kagome lattice might not be directly detected here, the observation of coexisting nonmagnetic sites, free spins, and frozen moments reveals an intrinsically inhomogeneous magnetism. Numerical simulations of a $\frac{1}{3}$-diluted kagome lattice rationalize this magnetic state in terms of a heterogeneous distribution of cluster sizes and morphologies near the site-percolation threshold.

We study the energy and the static spin structure factor of the ground state of the spin-1/2 quantum Heisenberg antiferromagnetic model on the kagome lattice. By the iterative application of a few Lanczos steps on accurate projected fermionic wave functions and the Green's function Monte Carlo technique, we find that a gapless (algebraic) U(1) Dirac spin liquid is competitive with previously proposed gapped (topological) Z2 spin liquids. By performing a finite-size extrapolation of the ground-state energy, we obtain an energy per site E/J=-0.4365(2), which is equal, within three error bars, to the estimates given by the density-matrix renormalization group (DMRG). Our estimate is obtained for a translationally invariant system, and, therefore, does not suffer from boundary effects, like in DMRG. Moreover, on finite toric clusters at the pure variational level, our energies are lower compared to those from DMRG calculations., Comment: Final published version. Main Paper (6 pages, 4 figures, 1 table) + Supplementary Material (4 pages, 8 figures)

Magnetic frustration helps destabilizing conventional Néel order at T = 0 in dimensions 2, and therefore allows the emergence of new original quantum phases. The S=1/2 Heisenberg Hamiltonian on the highly frustrated kagome lattice, which is made of corner-sharing triangles, is expected to stabilize such quantum states, including the spin liquid ones which do not break any symmetry even at T = 0. This thesis work focuses on the experimental study of two kinds of S=1/2 (Cu2+) kagome compounds using NMR and μSR local probes as well as thermodynamic measurements (magnetic susceptibility, specific heat).In Mg-herbertsmithite magnetic frustration occurs thanks to a first nearest-neighbor antiferromagnetic interaction J and is responsible for the spin liquid behavior observed down to T = J/10000. In comparison with the formerly known isostructural counterpart Zn-herbertsmithite, we showed that it shares similar physical magnetic properties while allowing sensitive structural refinements and therefore a control of the level of Cu/Mg substitutions defects. Our experiments performed on such well controlled materials allow us to investigate the origin of the dynamical relaxation in these compounds in relation with the existence of interplane spins defects. Kapellasite and haydeite possess both ferromagnetic (J1) and antiferromagnetic (Jd) interactions. They offer the possibility to explore the phase diagram generated by such competing interactions on the kagome lattice. For kapellasite, our μSR experiments evidenced a spin liquid character down to T ≈ J1/1000. We tracked the temperature dependence of the magnetic susceptibility probed by 35Cl-NMR as well as of the specific heat, from which the ratio Jd/J1 = 0.85 can be evaluated. This ratio locates the ground-state of kapellasite to be within an original non-coplanar spin phase described by 12 magnetic sublattices and called cuboc2. Magnetic exchanges in haydeite locate its ground-state within the ferromagnetic phase. Both our local and thermodynamic measurements point to a partial ferromagnetic transition at T = 4 K. This study confirms the relevance of the frustrated quantum kagome lattice to stabilize original quantum phases and suggests the existence of a new spin liquid phase, distinct from the one expected for antiferromagnetically coupled spins.; La déstabilisation de l’ordre antiferromagnétique de Néel au profit de nouvelles phases quantiques à température nulle à deux dimensions est envisageable grâce au phénomène de frustration magnétique. Le modèle théorique de spins Heisenberg S=1/2 répartis sur le réseau bidimensionnel frustré kagome, constitué de triangles joints uniquement par leurs sommets, est susceptible de stabiliser des phases quantiques originales de liquides de spin, qui ne présentent aucune brisure de symétrie à T = 0. Cette thèse a été consacrée à l’étude expérimentale de deux types de composés de spins S=1/2 (Cu2+) à géométrie kagome à l’aide de techniques spectroscopiques locales, la RMN et la μSR, ainsi que de mesures thermodynamiques (susceptibilité magnétique, chaleur spécifique). Dans Mg-herbertsmithite, la frustration est générée par une interaction d’échange premiers voisins antiferromagnétique J et est responsable d’un comportement liquide de spin jusqu’à des températures de l’ordre de J/10000. Par rapport au composé isostructural antérieur, Zn-herbertsmithite, nous avons montré qu’il possédait des propriétés physiques similaires tout en permettant une caractérisation fine du taux de défauts de substitutions Cu/Mg. Nos expériences réalisées à partir d’échantillons contrôlés permettent d’étudier finement l’origine des plateaux de relaxation observés en μSR à basse température en lien avec l’existence des défauts de spins interplans. La kapellasite et l’haydéite possèdent des interactions ferromagnétiques (J1) et antiferromagnétiques (Jd), offrant la possibilité d’explorer le diagramme de phases générées par la compétition de ces interactions sur le réseau kagome. Pour la kapellasite, nos mesures de μSR démontrent le caractère liquide de spin jusqu’à T ≈ J1/1000. La dépendance en température de la susceptibilité magnétique sondée par RMN du 35Cl ainsi que de la chaleur spécifique permettent d’évaluer le rapport Jd/J1 = 0.85, qui localise classiquement son fondamental au sein d’une phase originale de spins non coplanaires à 12 sous-réseaux appelée cuboc2. Les interactions présentes dans l’haydéite localisent son fondamental au sein de la phase ferromagnétique, en bon accord avec nos mesures qui indiquent une transition partielle à caractère ferromagnétique à T = 4 K. Cette étude confirme la pertinence du réseau kagome frustré pour la stabilisation de phases quantiques originales et démontre l’existence d’une nouvelle phase liquide de spin sur ce réseau, distincte de celle attendue pour des spins couplés antiferromagnétiquement.

Magnetic frustration helps destabilizing conventional Néel order at T = 0 in dimensions 2, and therefore allows the emergence of new original quantum phases. The S=1/2 Heisenberg Hamiltonian on the highly frustrated kagome lattice, which is made of corner-sharing triangles, is expected to stabilize such quantum states, including the spin liquid ones which do not break any symmetry even at T = 0. This thesis work focuses on the experimental study of two kinds of S=1/2 (Cu2+) kagome compounds using NMR and μSR local probes as well as thermodynamic measurements (magnetic susceptibility, specific heat).In Mg-herbertsmithite magnetic frustration occurs thanks to a first nearest-neighbor antiferromagnetic interaction J and is responsible for the spin liquid behavior observed down to T = J/10000. In comparison with the formerly known isostructural counterpart Zn-herbertsmithite, we showed that it shares similar physical magnetic properties while allowing sensitive structural refinements and therefore a control of the level of Cu/Mg substitutions defects. Our experiments performed on such well controlled materials allow us to investigate the origin of the dynamical relaxation in these compounds in relation with the existence of interplane spins defects. Kapellasite and haydeite possess both ferromagnetic (J1) and antiferromagnetic (Jd) interactions. They offer the possibility to explore the phase diagram generated by such competing interactions on the kagome lattice. For kapellasite, our μSR experiments evidenced a spin liquid character down to T ≈ J1/1000. We tracked the temperature dependence of the magnetic susceptibility probed by 35Cl-NMR as well as of the specific heat, from which the ratio Jd/J1 = 0.85 can be evaluated. This ratio locates the ground-state of kapellasite to be within an original non-coplanar spin phase described by 12 magnetic sublattices and called cuboc2. Magnetic exchanges in haydeite locate its ground-state within the ferromagnetic phase. Both our local and thermodynamic measurements point to a partial ferromagnetic transition at T = 4 K. This study confirms the relevance of the frustrated quantum kagome lattice to stabilize original quantum phases and suggests the existence of a new spin liquid phase, distinct from the one expected for antiferromagnetically coupled spins., La déstabilisation de l’ordre antiferromagnétique de Néel au profit de nouvelles phases quantiques à température nulle à deux dimensions est envisageable grâce au phénomène de frustration magnétique. Le modèle théorique de spins Heisenberg S=1/2 répartis sur le réseau bidimensionnel frustré kagome, constitué de triangles joints uniquement par leurs sommets, est susceptible de stabiliser des phases quantiques originales de liquides de spin, qui ne présentent aucune brisure de symétrie à T = 0. Cette thèse a été consacrée à l’étude expérimentale de deux types de composés de spins S=1/2 (Cu2+) à géométrie kagome à l’aide de techniques spectroscopiques locales, la RMN et la μSR, ainsi que de mesures thermodynamiques (susceptibilité magnétique, chaleur spécifique). Dans Mg-herbertsmithite, la frustration est générée par une interaction d’échange premiers voisins antiferromagnétique J et est responsable d’un comportement liquide de spin jusqu’à des températures de l’ordre de J/10000. Par rapport au composé isostructural antérieur, Zn-herbertsmithite, nous avons montré qu’il possédait des propriétés physiques similaires tout en permettant une caractérisation fine du taux de défauts de substitutions Cu/Mg. Nos expériences réalisées à partir d’échantillons contrôlés permettent d’étudier finement l’origine des plateaux de relaxation observés en μSR à basse température en lien avec l’existence des défauts de spins interplans. La kapellasite et l’haydéite possèdent des interactions ferromagnétiques (J1) et antiferromagnétiques (Jd), offrant la possibilité d’explorer le diagramme de phases générées par la compétition de ces interactions sur le réseau kagome. Pour la kapellasite, nos mesures de μSR démontrent le caractère liquide de spin jusqu’à T ≈ J1/1000. La dépendance en température de la susceptibilité magnétique sondée par RMN du 35Cl ainsi que de la chaleur spécifique permettent d’évaluer le rapport Jd/J1 = 0.85, qui localise classiquement son fondamental au sein d’une phase originale de spins non coplanaires à 12 sous-réseaux appelée cuboc2. Les interactions présentes dans l’haydéite localisent son fondamental au sein de la phase ferromagnétique, en bon accord avec nos mesures qui indiquent une transition partielle à caractère ferromagnétique à T = 4 K. Cette étude confirme la pertinence du réseau kagome frustré pour la stabilisation de phases quantiques originales et démontre l’existence d’une nouvelle phase liquide de spin sur ce réseau, distincte de celle attendue pour des spins couplés antiferromagnétiquement.

Volborthite Cu 3 V 2 O 7 ( OH ) 2 · 2 H 2 O is a natural frustrated antiferromagnet with a kagome-like magnetic lattice decorated by S = 1 2 spins. Our μ SR results confirm the existence of a dynamical plateau below 2 K, as initially demonstrated in [Phys. Rev. Lett. 91 (2003) 207603] despite sensible quantitative differences in the relaxation rate values. Thanks to additional low temperature SQUID and NMR data we show that this plateau occurs concurrently with a freezing transition. Moreover, we revisited the effect of diluting the magnetic lattice for small dilution rates between 1% and 15%. Comparison is made with the well studied kagome bilayer compounds. The ground state of volborthite appears to be far less stable with respect to dilution.

In this paper, we do a complete classification of valence-bond crystals (VBCs) on the kagome lattice based on general arguments of symmetry only and thus identify many new VBCs for different unit cell sizes. For the spin-1/2 Heisenberg antiferromagnet, we study the relative energetics of competing gapless spin liquids (SLs) and VBC phases within the class of Gutzwiller-projected fermionic wave functions using variational Monte Carlo techniques, hence implementing exactly the constraint of one fermion per site. By using a state-of-the-art optimization method, we conclusively show that the U(1) Dirac SL is remarkably stable towards dimerizing into all 6-, 12- and 36-site unit cell VBCs. This stability is also preserved on addition of a next-nearest-neighbor super-exchange coupling of both antiferromagnetic and ferromagnetic (FM) type. However, we find that a 36-site unit cell VBC is stabilized on addition of a very small next-nearest-neighbor FM super-exchange coupling, i.e. |J2|~0.045, and this VBC is the same in terms of space-group symmetry as that obtained in an effective quantum dimer model study. It breaks reflection symmetry, has a nontrivial flux pattern and is a strong dimerization of the uniform RVB SL., 16 pages, 25 figures. Invited paper for Focus issue on "Quantum Spin Liquids" of the New Journal of Physics