This thesis presents research of two novel magnetic materials, LiHoF4 and Tb2Ti2O7. Experiments were performed at low temperatures and in an applied magnetic field to study thermal expansion and magnetostriction using a capacitive dilatometer designed during this project. This thesis presents 3 distinct topics. This manuscript begins with a thermodynamic description of thermal expansion and magnetostriction. The design of a capacitive dilatometer suitable for use at ultra-low temperatures and in high magnetic fields is presented. The thermal expansion of oxygen free high conductivity copper is used as a test of the absolute accuracy of the dilatometer. The first material studied using this dilatometer was LiHoF4. Pure LiHoF4 is a dipolar coupled Ising ferromagnet and in an applied transverse magnetic field is a good representation of the transverse field Ising model. An ongoing discrepancy between theoretical and experimental work motivates further study of this textbook material. Presented here are thermal expansion and magnetostriction measurements of LiHoF4 in an applied transverse field. We find good agreement with existing experimental work. This suggests that there is some aspect of LiHoF4 or the effect of quantum mechanical fluctuations at finite temperatures which is not well understood. The second material studied is the spin liquid Tb2Ti2O7. Despite theoretical predictions that Tb2Ti2O7 will order at finite temperature, a large body of experimental evidence demonstrates that spins within Tb2Ti2O7 remain dynamic to the lowest temperatures studied. In addition Tb2Ti2O7 also exhibits anomalous thermal expansion below 20K, giant magnetostriction, and orders in an applied magnetic field. Thermal expansion and magnetostriction measurements of Tb2Ti2O7 are presented in applied longitudinal and transverse fields. Zero-field thermal expansion measurements do not repeat the previously observed anomalous thermal expansion. A large feature is observed in thermal expansion at 100mK, in rough agreement with existing experimental work. Longitudinal and transverse magnetic fields were applied to Tb2Ti2O7. Longitudinal magnetostriction measurements show qualitatively di erent behavior than previous observations. These measurements were taken along di erent crystal axes so direct comparison cannot be made. Thermal expansion measurements in an applied transverse field show evolution with the strength of the applied field. This evolution may relate to an ordering transition, however difficulties in repeatability in a transverse field require that these results be repeated in an improved setup.

Various experiments on Yb₂Ti₂O₇ have shown evidence of strange magnetic behaviour at low temperatures. Specific heat measurements on powder samples of Yb₂Ti₂O₇ show evidence of a sharp peak, indicating the occurence of a first order phase transition. Meanwhile, neutron scattering, Mössbauer absorption, and μSR measurements find no evidence of long range order below the temperature of this phase transition, leaving the nature of the low temperature phase a mystery. Quantifying the magnetic interactions in this material should allow us to better understand the low temperature behaviour of this material. In this study, we fit a symmetry allowed nearest-neighbour bilinear exchange model to quasi-elastic neutron scattering data collected well above the temperature of the experimentally observed phase transition. This neutron scattering data shows evidence of rods of scattering intensity along the ⟨111⟩ crystallographic directions. Neutron scattering probes the correlations between magnetic moments in a material, so fitting an interaction model to the neutron scattering is equivalent to fitting the interactions to the magnetic correlations. These correlations are driven by the interactions between the magnetic moments, so the neutron scattering should give us direct access to the form of these interactions. Using this method we successfully identify an anisotropic nearest-neighbour bilinear exchange model that reproduces the experimentally observed quasi-elastic neutron scattering. With this model we then proceed to compute real space correlation functions, finding that the rods of neutron scattering arise from the presence of strong correlations along nearest-neighbour chains. We also compute the bulk susceptibility and local susceptibility, obtaining very good fits to experiment with no variation of the model determined from the neutron scattering. The success of these calculations provides a further independent confirmation of the success of our interaction model in describing the magnetic interactions in Yb₂Ti₂O₇. Finally, we present a brief summary of ongoing work based on our anisotropic exchange model, including mean field calculations to determine the low temperature ground state of this model and classical Monte Carlo simulations to study the phase transition present in this model. We also discuss potential further studies of this and other models.

Complex magnetic oxides represent a unique intersection of immense technological importance and fascinating physical phenomena originating from interwoven structural, electronic and magnetic degrees of freedom. The resulting energetically close competing orders can be controllably selected through external fields. Competing interactions and disorder represent an additional opportunity to systematically manipulate the properties of pure magnetic systems, leading to frustration, glassiness, and other novel phenomena while finite sample dimension plays a similar role in systems with long-range cooperative effects or large correlation lengths. A rigorous understanding of these effects in strongly correlated oxides is key to manipulating their functionality and device performance, but remains a challenging task. In this dissertation, we examine a number of problems related to intrinsic and extrinsic low dimensionality, disorder, and competing interactions in magnetic oxides by applying a unique combination of standard magnetometry techniques and unconventional magnetocaloric effect and transverse susceptibility measurements. The influence of dimensionality and disorder on the nature and critical properties of phase transitions in manganites is illustrated in La0.7Ca0.3MnO3, in which both size reduction to the nanoscale and chemically-controlled quenched disorder are observed to induce a progressive weakening of the first-order nature of the transition, despite acting through the distinct mechanisms of surface effects and site dilution. In the second-order material La0.8Ca0.2MnO3, a strong magnetic field is found to drive the system toward its tricritical point as competition between exchange interactions in the inhomogeneous ground state is suppressed. In the presence of large phase separation stabilized by chemical disorder and long-range strain, dimensionality has a profound effect. With the systematic reduction of particle size in microscale-phase-separated (La, Pr, Ca)MnO3 we observe a disruption of the long-range glassy strains associated with the charge-ordered phase in the bulk, lowering the field and pressure threshold for charge-order melting and increasing the ferromagnetic volume fraction as particle size is decreased. The long-range charge-ordered phase becomes completely suppressed when the particle size falls below 100 nm. In contrast, low dimensionality in the geometrically frustrated pseudo-1D spin chain compound Ca3Co2O6 is intrinsic, arising from the crystal lattice. We establish a comprehensive phase diagram for this exotic system consistent with recent reports of an incommensurate ground state and identify new sub-features of the ferrimagnetic phase. When defects in the form of grain boundaries are incorporated into the system the low-temperature slow-dynamic state is weakened, and new crossover phenomena emerge in the spin relaxation behavior along with an increased distribution of relaxation times. The presence of both disorder and randomness leads to a spin-glass-like state, as observed in γFe2O3 hollow nanoparticles, where freezing of surface spins at low temperature generates an irreversible magnetization component and an associated exchange-biasing effect. Our results point to distinct dynamic behaviors on the inner and outer surfaces of the hollow structures. Overall, these studies yield new physical insights into the role of dimensionality and disorder in these complex oxide systems and highlight the sensitivity of their manifested magnetic ground states to extrinsic factors, leading in many cases to crossover behaviors where the balance between competing phases is altered, or to the emergence of entirely new magnetic phenomena.

The experimental quest for a quantum spin-liquid state (QSL) in frustrated magnetic systems addresses fundamental scientific interests, as this intriguing quantum phase provides excellent grounds for discovering exotic collective phenomena. ZnCu3(OH)6Cl2 (herbertsmithite), an S=1/2 kagomé-lattice Heisenberg antiferromagnet, is the most promising candidate for experimentally realizing a QSL. However, despite years of intense research, the nature of its paramagnetic ground state remains highly debated. The root cause of the controversy lies in the difficulty in distinguishing the effects of defects from the intrinsic properties of the kagomé lattice. In this thesis, we present 17-O nuclear magnetic resonance (NMR) measurements of an isotope-enriched ZnCu3(OH)6Cl2 single crystal. We succeeded in distinguishing the intrinsic magnetic behavior of the kagomé lattice from the defect-induced phenomena down to T~0.01J, where J~200K is the Cu-Cu super-exchange interaction. We identify NMR signals arising from the nearest-neighbor 17-O sites of Cu2+ defects occupying the Zn2+ interlayer sites. From the 17-O Knight shift measurements, we show that these Cu2+ defects induce a large Curie-Weiss contribution to the bulk-averaged susceptibility at low temperatures. Moreover, our 17-O single-crystal lineshapes show no signature of nonmagnetic Zn2+ defects within the kagomé lattice, and therefore, we rule out “anti-site disorder” as a cause of the paramagnetic ground state in ZnCu3(OH)6Cl2. Most importantly, we demonstrate that the intrinsic spin susceptibility of the kagome lattice asymptotically tends to zero below T~0.03J, indicating the presence of a finite gap Δ = 0.03~ 0.07J in the spin excitation spectrum; this gap is completely suppressed under the application of a high magnetic field of ~ 9T. The behavior of low-energy spin fluctuations probed by the 17-O nuclear spin-lattice relaxation rate is consistent with the gap signature observed for the 17-O Knight shift. In short, our 17-O NMR results provide the first experimental evidence for a gapped QSL realized in ZnCu3(OH)6Cl2. / Thesis / Doctor of Philosophy (PhD)

<p>We report low temperature specific heat and positive muon spin rotation measurements of both polycrystal and single crystal Yb2Ti2O7. Our zero field (ZF) measurements show little spin relaxation temperature dependence in the polycrystal Yb2Ti2O7, contrast to previously reported results. We observe no collinear ferromagnetic order, rather a hidden order ground state where spin fluctuations remain dynamic down to 16 mK. Single crystal Yb2Ti2O7 zero field measurements with the crystallographic [111] direction parallel to the initial muon polarization show small but measurable temperature dependence. In addition, our transverse field measurements show the spin susceptibility undergoes a distinct change at temperatures corresponding to the magnetic transition measured in the specific heat.</p> / Master of Science (MSc)

In the study of strongly interacting condensed-matter systems controlled microscopic theories hold a key position. Spin-wave theory, large-N expansion, and $epsilon$-expansion are some of the few successful cornerstones. In this doctoral thesis work, we have developed a novel large-$d$ expansion method, $d$ being the spatial dimension, to study model Hamiltonians hosting a quantum phase transition between a paramagnet and a magnetically ordered phase. A highlight of this technique is that it can consistently describe the entire phase diagram of the above mentioned models, including the quantum critical point. Note that most analytical techniques either efficiently describe only one of the phases or suffer from divergences near the critical point. The idea of large-$d$ formalism is that in this limit, non-local fluctuations become unimportant and that a suitable product state delivers exact expectation values for local observables, with corrections being suppressed in powers of $1/d$. It turns out that, due to momentum summation properties of the interaction structure factor, all diagrams are suppressed in powers of $1/d$ leading to an analytic expansion. We have demonstrated this method in two important systems namely, the coupled-dimer magnets and the transverse-field Ising model. Coupled-dimer magnets are Heisenberg spin systems with two spins, coupled by intra-dimer antiferromagnetic interaction, per crystallographic unit cell (dimer). In turn, spins from neighboring dimers interact via some inter-dimer interaction. A quantum paramagnet is realized for a dominant intra-dimer interaction, while a magnetically ordered phase exists for a dominant (or of the same order as intra-dimer interaction) inter-dimer interaction. These two phases are connected by a quantum phase transition, which is in the Heisenberg O(3) universality class. Microscopic analytical theories to study such systems have been restricted to either only one of the phases or involve uncontrolled approximations. Using a non-linear bond-operator theory for spins with S=$1/2$, we have calculated the $1/d$ expansion of static and dynamic observables for coupled dimers on a hypercubic lattice at zero temperature. Analyticity of the $1/d$ expansion, even at the critical point, is ensured by correctly identifying suitable observables using the mean-field critical exponents. This method yields gapless excitation modes in the continuous symmetry broken phase, as required by Goldstone\'s theorem. In appropriate limits, our results match with perturbation expansion in small ratio of inter-dimer and intra-dimer coupling, performed using continuous unitary transformations, as well as the spin-wave theory for spin-$1/2$ in arbitrary dimensions. We also discuss the Brueckner approach, which relies on small quasiparticle density, and derive the same $1/d$ expansion for the dispersion relation in the disordered phase. Another success of our work is in describing the amplitude (Higgs) mode in coupled-dimer magnets. Our novel method establishes the popular bond-operator theory as a controlled approach. In $d=2$, the results from our calculations are in qualitative agreement with the quantum Monte Carlo study of the square-lattice bilayer Heisenberg AF spin-$1/2$ model. In particular, our results are useful to identify the amplitude (Higgs) mode in the QMC data. The ideas of large-$d$ are also successfully applied to the transverse-field Ising model on a hypercubic lattice. Similar to bond operators, we have introduced auxiliary Bosonsic operators to set up our method in this case. We have also discussed briefly the bilayer Kitaev model, constructed by antiferromagnetically coupling two layers of the Kitaev model on a honeycomb lattice. In this case, we investigate the dimer quantum paramagnetic phase, realized in the strong inter-layer coupling limit. Using bond-operator theory, we calculate the mode dispersion in this phase, within the harmonic approximation. We also conjecture a zero-temperature phase diagram for this model.

Cette thèse porte sur l'étude d'oxydes d'iridium dont le fort couplage spin-orbite est susceptible de générer de nouvelles phases électroniques et magnétiques. Deux familles de composés ont été considérées: Sr3MM’O6, à chaines de spins mixtes arrangées sur réseau triangulaire, et R2Ir2O7, à réseaux pyrochlores interpénétrés de spins. Ils ont été synthétisés sous forme polycristalline et pour certains sous forme monocristalline puis étudiés macroscopiquement par mesure d'aimantation. Ils ont ensuite été sondés microscopiquement par diffusion de neutrons et de rayons X. Nos mesures montrent que dans les composés à chaînes de spins Sr3NiPtO6 et Sr3NiIrO6 les ions Ni2+ présentent une très forte anisotropie magnétocristalline planaire perpendiculaire à l'axe des chaînes. Nous démontrons que ceci stabilise dans Sr3NiPtO6 une phase non magnétique dite « large-D ». Cette anisotropie se manifeste dans Sr3NiIrO6 à haute température. Ce composé s'ordonne cependant à basse température dans une structure magnétique avec les moments alignés le long de l'axe des chaînes. Nous expliquons ce changement d'anisotropie comme étant dû à la présence des ions Ir4+ dont le couplage spin-orbite produit une forte anisotropie des interactions Ni-Ir qui confinent les moments magnétiques le long des chaînes. Concernant les pyrochlores iridates R2Ir2O7, les mesures d'aimantation et de diffraction de neutron sont cohérents avec un ordre "all-in/all-out" des moments magnétiques des ions Ir4+, révélé indirectement via le comportement du sous-réseau des terres rares R. Cet ordre est le seul compatible avec la phase semi-métal de Weyl prédite comme résultant du fort couplage spin-orbite. Le comportement du sous-réseau de terre rare R dépend de l'anisotropie magnétocrystalline des ions R3+. Les ions à anisotropie uniaxiale locale sont polarisés par le champ moléculaire produit par l'ordre de l'iridium dont la direction coïncide avec l'axe d'anisotropie. Les ions à anisotropie locale planaire perpendiculaire à cette direction ne présentent pas d’ordre magnétique induit par celui de l'iridium. A plus basse température, les interactions entre terres rares génèrent des comportements magnétiques plus complexes. / This thesis focuses on the study of iridium oxides, in particular on the consequences of the strong spin-orbit coupling of the iridium. Two families of compounds have been investigated: Sr3MM’O6, with mixed spin chains arranged on a triangular lattice, and R2Ir2O7 with interpenetrated pyrochlores networks of spins. Polycrystalline samples have been synthetized and in some instances single crystals were successfully grown. They were investigated macroscopically by magnetization measurements and probed microscopically by neutron and synchrotoron X-ray scattering experiments. Our measurements showed that in the spin chain compounds Sr3NiPtO6 and Sr3NiIrO6 the Ni2+ ions show a strong easy plane magnetocrystalline anisotropy, perpendicular to the chain axis. This stabilizes in Sr3NiPtO6 the so-called "large-D" non-magnetic phase. The planar anisotropy comes out in Sr3NiIrO6 at high temperature. The compound however orders at low temperature in a magnetic configuration with all the magnetic moments confined along the chain axis. We explain this change of anisotropy as due to the Ir4+ ions whose spin-orbit coupling produces a strong anisotropy of the intra-chain Ni-Ir magnetic interactions overwhelming the single-ion Ni2+ anisotropy. Concerning the pyrochlore iridates R2Ir2O7, magnetization measurements and neutron powder diffraction experiments are consistent with an "all-in/all-out" magnetic ordering of the Ir magnetic moments, revealed indirectly through the magnetic behavior of the rare-earth sublattice. This ordering is the only one consistent with a Weyl semi-metal phase predicted to arise from the spin-orbit coupling. The magnetic behavior of the rare-earth sublattice depends on the rare earth magnetocrystalline anisotropy. The ions with local uniaxial anisotropy are polarized by the Ir molecular field, whose direction coincides with the anisotropy axis. The ions with local planar anisotropy perpendicular to this direction show no iridium induced long-range magnetic ordering. At lower temperature, rare-earth interactions generate more complex magnetic behaviors.

La frustration – c'est-à-dire la présence d'interactions de nature compétitive – donne lieu à des effets de grande complexité en physique. La glace – aussi bien la phase bien connue de l'eau, que ses analogues magnétiques, dites « glaces de spin » – en offre un exemple remarquable. Pour des interactions à courte portée et des degrés de liberté classiques, son état fondamental est infiniment dégénéré, et comporte en outre des corrélations à longue portée induites par une contrainte locale, caractéristiques de la phase dite de Coulomb. Ses excitations élémentaires correspondent au retournement d'un dipôle qui se « fractionnalise » en deux monopôles. Dans cette thèse nous nous intéressons à la stabilité de cette phase de Coulomb dans la glace bidimensionnelle – réalisée aussi bien comme glace de protons dans des composés organiques, que comme glace de spin dans des systèmes nanomagnétiques. Dans le cas classique, les interactions dipolaires – présentes dans les systèmes expérimentaux – déstabilisent la phase de Coulomb dans son état fondamental. Cependant, une déformation de la simple géométrie planaire permet de récupérer cette phase dans un régime où différents états ordonnés entrent en compétition. Dans le cas quantique, les fluctuations dues à un champ magnétique transverse induisent une brisure de symétrie dans l'état fondamental qui, à basse température, cède la place à une phase de Coulomb quantique, réalisant un liquide de spin quantique avec excitations fractionnalisées. Nos résultats sont obtenus à l'aide de méthodes analytiques (analyse harmonique classique et quantique et théorie de perturbations) aussi bien que numériques (Monte Carlo) fondées sur des algorithmes originaux. / Frustration – namely the presence of competing interactions – gives rise to highly complex effects in physics. Ice – be it the well known water ice or its magnetic equivalent, the so-called spin-ice – offers a remarkable example in this context. For short-range interactions and classical degrees of freedom, its ground state is infinitely degenerate, and exhibits long-range correlations induced by a local constraint, characterizing the so-called Coulomb phase. Its elementary excitations correspond to the flip of a dipole “fractionalizing” into two monopoles. In this thesis we are interested in the stability of this Coulomb phase in the two-dimensional ice – realized in the form of a proton ice in organic compounds as well as of spin ice in nanomagnetic systems. In the classical case, dipolar interactions – present in the experimental systems – destabilize the Coulomb phase in the ground state. However, a slight deformation of the simple planar geometry allows to recover this phase in a regime where different ordered states compete with each other. In the quantum case, fluctuations due to a transverse magnetic field induce a symmetry breaking in the ground state, that melts at low temperature into a quantum Coulomb phase, realizing a quantum spin liquid with fractionalized excitations. Our results were obtained with analytical (classical and quantum normal-mode analysis and perturbation theory) as well as numerical techniques (Monte Carlo) based on original algorithms.

Les systèmes multiferroïques magnéto-électriques sont des matériaux multifonctionnels très importants du point de vue des applications dans le domaine de l’électronique ou de la spintronique puisqu’ils présentent simultanément des ordres électriques et magnétiques généralement couplés. Ils peuvent donc répondre à la fois à l’application d’un champ magnétique et d’un champ électrique. L’une des familles de multiferroïques magnéto-électriques les plus étudiées est la série RMn₂O₅ où R est une terre rare. Ma thèse porte sur le rôle de la terre rare sur le caractère multiferroïque et sur les propriétés magnétiques de ces composés. De nombreux travaux ont déjà été publiés mais très peu concernent les composés SmMn₂O₅ et GdMn₂O₅, difficiles à étudier aux neutrons. Pourtant ce sont des composés clés, à la frontière entre des composés à terres rares légères qui ne sont pas multiferroïques et ceux à terres rares lourdes qui le sont. Je me suis donc intéressé à ces composés. Grâce à une étude théorique basée sur une analyse de symétrie et des calculs numériques ab initio tenant compte du fort couplage spin-orbite, nous avons pu prévoir un modèle pour l’ordre magnétique stabilisé dans Sm et Gd. Nous avons en parallèle étudié expérimentalement les structures magnétiques pour ces deux composés par le biais d’une analyse des donnés de diffraction de neutrons sur poudre utilisant des matériaux isotopes de Sm et Gd. Ma thèse a permis d’une part de valider le mécanisme d’échange striction comme origine du couplage magnéto-électrique dans cette série importante de multiferroïques. Elle a permis d’autre part de mettre en évidence l’existence d’une interaction d’échange supplémentaire dans GdMn₂O₅, à l’origine de la forte polarisation électrique dans ce membre de la série. Ces résultats amènent plus de clarté à la compréhension de la multiferroïcité dans ces systèmes. / The magneto-electric multiferroic systems are multifunctional materials very important for applications in the field of electronics or spintronics since they present simultaneously electrical and magnetic orders, which are generally coupled. They can thus respond to both the application of a magnetic field and an electric field. One of the most studied magneto-electric multiferroic families is the RMn₂O₅ series where R is a rare earth. My thesis deals with the role of rare earth on the multiferroic and the magnetic properties of these compounds. Numerous works have already been published, but very few concern the compounds SmMn₂O₅ and GdMn₂O₅, difficult to study with neutrons. Yet these are key compounds, on the border between light rare earth compounds that are not multiferroic and heavy rare earths that are. This explains my interest for these compounds. Using a theoretical study based on a symmetry analysis and numerical calculations ab initio taking into account the strong spin-orbit coupling, we were able to predict a model for the stabilized magnetic order in Sm and Gd. We have experimentally studied the magnetic structures for these two compounds by means of an analysis of the powder neutron diffraction data using Sm and Gd isotopes. On one hand, my thesis allowed to validate the mechanism of exchange striction as origin of the magnetoelectric coupling in this important series of multiferroics. On the other hand, it has made it possible to demonstrate the existence of an additional exchange interaction in GdMn₂O₅, at the origin of the strong electrical polarization in this member of the series. These results provide greater clarity to the understanding of multiferroicity in these systems.

Quantum spin liquids represent a novel phase of magnetic matter where quantum fluctuations are large enough to suppress the formation of local order parameters, even down to zero temperature. Quantum spin liquid states can emerge from frustrated quantum magnets. These states show several peculiar properties, such as topological order, fractional excitations, and long-range entanglement. The Kitaev spin model on the honeycomb lattice is one of the few models proposed which can exactly show the existence of a $\mathbb{Z}_2$ quantum spin liquid. The model describes spins featuring frustrated compass interactions, and it exhibits a quantum spin liquid ground state. The model's ground state can be found exactly by representing spins in terms of Majorana fermions. It turns out that spin excitations fractionalize into two degrees of freedom: spinless matter fermions and flux excitations of the emergent $\mathbb{Z}_2$ gauge theory. Recently, possible solid-state realizations of Kitaev quantum spin liquids have been proposed in a class of frustrated Mott insulators. Unfortunately, experiments can not unambiguously identify quantum spin liquids, due to their elusive nature. Nevertheless, indirect observations on a spin liquid state can be done by looking at its excitations. Along this line, thermal transport investigations provide for an option to study heat-carrying excitations, and thus the properties of the related spin liquid state. In this doctoral thesis work, I performed a study of longitudinal thermal transport properties in the two-dimensional Kitaev spin model. This study aims to advance the understanding of transport in prototypical frustrated quantum magnets that might harbor Kitaev physics, and in particular quantum spin liquid states. For this purpose, I explored the model for varying exchange coupling regimes $-$ to underline the impact of anisotropy on transport $-$ and I studied transport over a wide range of temperatures. Transport properties have been explored within the formalism of the linear response theory. Based on the latter, thermal transport coefficients can be evaluated by calculating dynamical energy-current auto-correlation functions. First, I performed an analytical study of the uniform gauge sector of the model $-$ where excitations of gauge degrees of freedom are neglected. Analytical findings for the energy-current correlations, and their related transport coefficients, imply a finite-temperature ballistic heat conductor in terms of free matter fermion excitations $-$ independent of exchange couplings. Second, thermal transport has been studied at finite temperatures, considering thermal gauge excitations off the uniform gauge sector. For this purpose, I made use of two complementary numerical methods able to treat finite-temperature systems. On the one hand, I resorted on the exact diagonalization of the Kitaev Hamiltonian given in terms of fermions and a real-space dependent $\mathbb{Z}_2$ gauge potential, to study relatively small systems. On the other hand, I used an approximate method based on a mean-field treatment of thermal gauge fluctuations. The method allowed to extend the study of thermal transport to systems with up to $\sim\mathcal{O}(10^4)$ spinful sites. It made possible the computation of correlation functions by reducing the exact trace over all gauge states to an average over dominant gauge states suited to a given temperature range. The reliability of the method has been checked by comparing to numerically exact thermodynamics of systems. Based on the thermodynamic analysis, the method has been restricted to a temperature range where the mean-field treatment of gauge fluctuations is acceptable. Within such temperature range, the method succeeded in well reproducing exact results. The prime advantage of this method is its capability to reveal important features in the energy-current correlation spectra, not captured by the exact diagonalization approach because of finite-size effects. I found that the energy-current correlation spectra, in the presence of thermal gauge excitations, show clear signatures of spin fractionalization. In particular, the low-energy part of spectra displays features arising from a temperature-dependent matter-fermion density relaxation off an emergent thermal gauge disorder. This static gauge disorder also leads to the appearance of a pseudogap in the zero-frequency limit, which closes in the thermodynamic limit. The extracted dc heat conductivity is consequently influenced by this interplay between matter fermions and gauge degrees of freedom. The anisotropy in the exchange couplings moves Kitaev systems through gapless and gapped phases of the matter fermion sector. Effects of anisotropy are visible in the dc conductivities which display a low-temperature dependence crossing over from power-law to exponentially activated behavior upon entering the gapped phase. Therefore, I found that in the thermodynamic limit, two-dimensional Kitaev systems feature dissipative transport, regardless of exchange couplings. This finding is in contrast to the ballistic transport found discarding gauge excitations in the uniform gauge sector, which underlines the relevance of gauge degrees of freedom in thermal transport properties of Kitaev systems.