 Thank you. It's great pleasure for me to be here. And I'd like to thank the organizers for inviting me to this summer school. My name is Yuichi Kasahara from Kyoto University, Japan. So today, I'd like to talk about emergent quasi-particle excitations in a canton spin liquid and how to probe these emergent quasi-particle by summer transport experiments. So since I'm an experimentist, so I don't go much about the details of theory. But anyway, I will talk about the recent progress in the experiment on the spin liquid. So this is the outline of my talk. So first, since the lecture for the canton spin liquid was yesterday, so let me start with a very brief introduction of canton spin liquids. Then I discussed the summer transport in the canton spin liquid state, and especially in organic materials with a triangular lattice. So then in the afternoon, I will review the two recent experimental progress. So an observation of photons and monopoles in a pyroclon magnet, and the observation of myelona fermions in honeycomb magnets. As you will know, there are three state of matters, gas, liquid, and solid. And in classical systems, the atoms are frozen at the absolute zero temperature, and they usually crystallized into lattice with the ordering in conventional manner. So however, when the canton fluctuations become important, when the canton fluctuations at Heisenberger society become very important, this canton fluctuation prevent the conventional ordering, and the system become into the canton liquid state. So the most well-known example of the canton liquid is a liquid film, as you may know. The canton spin liquid is a spin analog of the canton liquid, in which a long-range order is destroyed. So strong canton fluctuations met a long-range order, such as a pyromagnetic order or antiretron order, even at zero temperature. So in this state, the ground state is characterized by the massive entanglement of local spins. So the notion of a canton spin liquid is well-established in one-dimensional systems. For one-dimensional Heisenberg antiretron magnetic chain, the ground state is a spin liquid. And actually, Tomonaga-Rachtinger liquid with spin correlation function shows the power law with respect to the distance. So this correlation is called the algebraic spin correlation. Then elemental excitation in this canton spin liquid is given by the spin flip of the one spin, which creates a pair of the domain here, so which is called the spinon. So this spinon carries a spin hub, but does not carry an electric charge. So this spinon is a gapless and propagates like this way, without any energy costs. Because these states are energetically equivalent. So this spinon excitation has been confirmed experimentally by an elastic neutron scattering as shown here. So this is a theoretical numerical result. And this is an experimental result. As you can see, these two are very good. So much very well. The most important to me, continuous excitation, magnetic continuum, appears at the gamma point in a wide range of energy. So this is a, this provides some signature of the spinon excitations. However, the situation is much more complicated in two and three dimensions. So it is widely believed that the geometrical frustration is necessary for realizing the canton spin liquid state in these two or three dimension systems. So the well known examples of the geometrically frustrated lattice include the two-dimensional triangular lattice, triangular and Kagome lattices, and three-dimensional pi-lock lower lattices. So although spin liquid is a difficult capture, but it is with several exotic physical properties, such as topological order, fluxionalized excitation, such as the spinon, and the gauge fluctuations. So next, let me discuss the two-dimensional, sorry, canton spin liquid in the two-dimensional triangular lattice. So there are possible ground states in the two-dimensional triangular antipiromagnets. So in conventional magnet, in classical magnet, so the ground state will be a nail order with 100 degrees structure as shown here. So this may be possible even if the canton fluctuations are present. So the next possibility is the balance point solid, so in which the singlet align periodically, so forming the long range order of spin singlet. So sorry, in nail state, both spin rotational symmetry and lattice symmetry is broken, but in this balance bond state, spin rotational symmetry, it's preserved, but lattice symmetry is broken because of this alignment. So then the final possibility is a spin-liquid state in which there is no simple symmetry breaking, including a spin rotational lattice symmetry is preserved. So because of this reason, the spin-liquid state is a very difficult capture. So the canton spin-liquid state of the Heisenberg, so canton spin-liquid state of two-dimensional triangular lattices in the Heisenberg model has been proposed by Fadakas and Andersson by considering the resonance balance bond state. In this case, the resonance between the different conflation superpositions as shown here. So the resonance between these states leads to the spin-like wave function, so leading to the canton spin-liquid state. However, it has been shown that even in the presence of a canton fluctuation, conventional long range order with this 130 degrees structure is more energetically preferable. So therefore, to achieve canton spin-liquid state in Heisenberg triangular lattices, we need more further frustrations. Recently, it has been shown that the ring exchange model as shown here provide further frustration in this kind of triangular systems. So I don't tell you about the detail, but this ring exchange interaction becomes important when the system is a motor insulator. And so the system is near the insulator to metal transition. So among real material systems, the organic compounds has been extensively studied for possible canton spin-liquid in the triangular lattices. So here, I'd like to show two examples. So copper et compounds and paradigm-dimit compounds, so in which the et molecules and PD-dimit molecules form the two-dimensional layer. So these two-dimensional layers are separated by anion or cation layers. These anion-cation layers are non-magnetic. So although the system, the original system is a quarter field, but these molecules form the dimer as shown here. Then leading to the, so giving rise to the half field, a more insulating state. So then half spins reside on this dimer, then forming the triangular lattice as shown here. In both compounds, J is relatively high. As large as the 200, it's close to the room temperature. So yes. So here I showed the result of NMR spectrum for the spin-liquid candidate material and the material which shows the anti-ferromagnetic order at low temperature. So as you can see in the anti-ferromagnet, this spectrum shows the splitting and the broadening at low temperature because of the magnetic ordering. In contrast, this spin-liquid compound, the spectrum do not show any splitting and broadening indicating the absence of internal field and magnetic order down to the very low temperature, so certain times smaller than the exchange interaction. This, the situation is similar in this compound. So no broadening in the NMR spectrum. And also via the Minnesota measurement reported the absence of internal magnetic field down to the very low temperature. So I'd like to emphasize that in this system, by changing the cations, so we can tune the system from the anti-ferromagnetic ordered state to the quantum spin-liquid state and to the charge-ordered state. So I will show later. So this charge-ordered material can be used as a reference for some conductivity measurements. So then for this two-dimensional triangular Heisenberg magnets, there are several theoretical proposals that have been reported, including resonating valence from the liquid, as I mentioned before. And the Keiler spin-liquid, that spontaneously breaks the time reversal symmetry. Kantan dimer-liquid, Jesus spin-liquid, something like that, and so on. So to establish the nature of Kantan spin-liquid, so it is very important to identify the elemental excitation. So because such excitation is very related to the nature of Kantan spin-liquid. However, it's very difficult to identify elemental excitations. So several possibilities have been several excitations and proposals, including the spin-on with Fermi surface by Zon-Meyer and Fermion. So they have not been established. Most of them have not been established so far. So next, let me discuss some transport properties in organic insulator with triangular lattices. So the question is to identify the elemental excitation and the nature of spin-liquid. So the question is how to probe the elemental excitation experimentally. So neutron scattering is still the powerful probe for detecting such excitations, as I mentioned before. Because, in fact, in two-dimensional Kagome, Harvard-Smithite, so magnetic continuum similar to the one-dimensional system has been observed, indicating the presence of spinons. However, the energy resolution of neutron scattering is very limited. So it's difficult to study much lower energy scales. And also, in two or three-dimensional systems, so it depends on the materials. But some, so Magnon or some other excitation can produce this kind of magnetic continuum. So therefore, interpretation of such continuum is very controversial. And the next year, so specific heat, and sorry, this is a specific heat, and NMR is a very sensitive probe for the very low energy excitations. So actually, in the triangular system, the gapless excitation has been reported by a specific heat measurement and the NMR measurement by the observation of a so residual linear term in the specific heat and the power law temperature dependence of the NMR relaxation rate. However, these quantities sometimes contain the Schottky contribution in specific heat and impurity contribution in the NMR measurement. So therefore, we need a more experimental probe for examining the elemental excitations. So here, I mainly discussed the thermal transport properties, such as longitudinal thermal conductivity and the thermal whole conductivity. So the sample is attached to the thermal bus and connected to the thermometer and the heater. Then so we can generate, if you inject the thermal current here, so the sample is heated up here and produce the temperature gradient along this direction. So then when you apply the magnetic field perpendicular to this plane, so the transverse thermal gradient can appear. So actually, the thermal transport coefficient has an advantage compared to the specific heat and NMR because they can sensitively probe the low energy itinerant excitations. And these quantities do not affect it by localized impurities and are not contaminated by Schottky contributions. So I will show you the later the quantum spin leakage transport heat very well. So that's the thermal transport. It's a very powerful technique for examining the quantum spin leakage. So let me show one example. Actually, so thermal conductivity measurement in one dimensional Heisenberg chain. So generally, thermal conductivity is given by this equation. So this is given by the product of the heat capacity and the velocity of the quasi-particles and the mean irreversible of the quasi-particles. However, in real material systems, there are several types of heat carriers, including electrons, phonons, magnons, and so on. So in magnetic insulator with no charge degree of freedom, there is no electronic contribution but so contribution. So however, usually, phonon contribution is very significant, especially at the finite temperature. So therefore, it is difficult to extract the phonon contribution and the spin contributions. However, so in one dimensional systems, this can be separated. These two can be separated by measuring the thermal conductivity along the chain direction and perpendicular to the chain directions. So these are the examples of the real material of the one-dimensional systems, this one and this one. So as you can see, clear difference can be seen in the thermal conductivity along the chain and perpendicular chain for both components. So assuming the phonon contribution to these directions are same, this difference gives the spin contribution to the thermal conductivity. So then in both components, so spin contribution develops very high temperatures, and it has been shown that the gapless itinerant spin excitation is present in the system. So this is consistent with the expectation for the spinon propagation. So let's move on to the two-dimensional organic materials. So this is a result for the copper ET compounds. Here shows a specific heat and thermal conductivity as a function of temperature. So this is a spin-digit materials. This black one is for the magnetically ordered materials. As you can see, so in specific heat contains both localized and itinerant contributions, but thermal conductivity only have itinerant contributions. So in specific heat shows a finite residual term in the linear temperature term. So specific heat divided by temperature, C over T. So the presence of finite C over T is very similar to the metal. And this gamma value is as large as 13 Kelvin. So this indicates that the presence of a gapless and fermionic excitation at low temperature. So on the other hand, thermal conductivity so it's difficult to say from this figure. Thermal conductivity shows the activation type temperature dependence. And we go to approach 0 with decreasing temperature. So this indicates that spin excitation is a gap at low temperature. Here I'd like to stress that the thermal conductivity in non-magnetic material is much lower than this compound. So in implying that the phonon contribution is, so this is not from the phonon contribution, but from the spin contributions. So combining these results, these results indicates that the system is highly inhomogeneous. And the gapless excitation originated from the localized contribution. So in fact, NMR and MIASR measurement suggests that the highly inhomogeneous grand state in this material. This shows the stretched exponent of NMR relaxation curve. And if this exponent is 1, the system is inhomogeneous. But if it becomes lower than the 1, so this indicates the system is highly inhomogeneous. As you can see, this exponent in this material is much lower than half. So this is consistent with the observation result of thermal conductivity and specific heat. So then the question is what is occurring in the inhomogeneous system? So as you can see here, so this material, paradigm limit compounds, the stretched exponent is close to 1. So indicating the inhomogeneous content spin leakage state in this compound. So I'd like to move on to this material. So here I show the specific heat as a function of temperature for spin leakage material and charge ordered material. So here the cation is different. So as you can see, finite C over T is observed only in the quantum spin leakage materials, indicating the presence of a gap-less fermionic excitations in the ground state. However, we should be careful because the specific heat contains the large short key contribution at low temperature. So this data, such contributions are already subtracted. So this conclusion might be very dangerous. So it depends on how to subtract short key contribution. So the thermal conductivity has been measured. So since, again, thermal conductivity do not, in thermal conductivity, thermal conductivity does not contaminate it by, not contaminated by the short key contributions. As you can see, the finite thermal conductivity is observed only in the quantum spin leakage compound. So it is consistent with the specific heat. Then combining with the specific heat, we, so both finite residual term in thermal conductivity and the specific heat indicates that the presence of a gap-less itinerant excitations with a fermionic character. As I mentioned before, thermal conductivity is related to this equation. So we can estimate the mean-funny path of a quasi-particle using the specific heat, the observed linear term in the specific heat. This value, and they are using the b of a quasi-particle from the dispersion relation of the b of a quasi-particle by assuming the linear dispersion of the spinons using the exchange interaction energy. Then we obtain the mean-funny path of spinons is as large as 0.5 micrometer, which is much, much larger than 500 times longer than the interspin distance. So indicating the highly mobile quasi-particle excitations. So this kind of very long range, as this indicates, the spin correlation has kept a very long distance, which is consistent with the algebraic correlation in the expected one-dimensional system and for the two-dimensional system. So finally, let me tell you that the observed thermal conductivity is actually very large, so which is comparable to the fact observed in metals. Here I draw the line for the brass. So this observed value is the same order as in the brass, so metals. So this actually showed that the significance of this kind of very large thermal conductivity. And OK. So here, this highly mobile spin excitation may provide constraints on the theory, but it's very difficult to identify the nature of quantum spin leakage. So next, let me discuss the thermal hole effect. The thermal hole effect is the thermal analog of the hole effect. So here, OK. So usually, in isotropic Heisenberg magnets, such a thermal hole effect is expected to be 0. However, in the presence of Jalsinski-Morilla interaction, asymmetric spin orbit interaction, this induces spin chirality of finite spin chirality, giving rise to the fictitious magnetic field. So this fictitious magnetic field can act as a low-range force over to the quasi-particles, leading to the finite thermal conductivity in the system, even in the magnetic insulator. There's no charge with no charge degree of freedom. So in fact, thermal hole effect has been proposed in the insulating anti-film magnetic insulator due to magnet hole effect arising from very curvature. And the spin hole effect in quantum spin leakage state is a spin-on film surface. The bosonic spinons with U1 spin leakage these are proposed in magnetic insulators. So OK. So experimentally, such hole effect has been actually observed in anti-film magnetically-ordered system. Magnum hole effect has been reported in this compare. Actually, thermal hole effect is 0 at high temperature, but it appears below the Q-lead temperature. In this material, thermal hole effect appears, it's observed, but it appears at a very high temperature above the characteristic temperature of the exchange interaction, indicating the thermal hole effect in the paramagnetic state. At present stage, there's no theory for the paramagnetic state. So the nature of the thermal hole effect in this material still controversial. So there are only a few reports for the thermal hole effect in the spin leakage state, but recently reported in the Kagome-Volvosite and Kagome-Kaperosite. Here shows the result of Kagome-Volvosite. So in this material, Kaper-Atoms forms the two-dimensional Kagome lattices, and it starts layer by layer as shown here. So this is a result for thermal hole conductivity as a function temperature. Here, vertical axis are normalized by temperature divided by temperature and magnetic field. So I would like to note that this material shows the anti-pilomagnetic order at about 1 Kelvin, but the exchange energy is about 60 Kelvin. So anti-pilomagnetic order is strongly suppressed, and above male temperature, spin leakage state is realized. So as you can see, in the thermal hole effect, it's absent at high temperature, but appears below this 60 Kelvin, where spin correlation become important. So when a complex wall appears upon entering the spin leakage state, with decreasing temperature, it shows a peak, and then shows a sign change below the male temperature. So this suggests the nature of the hole effect in the order state and the spin leakage states are different. And more importantly, the final thermal hole effect appears in the spin leakage state. And this peak appears at a temperature where magnetic sensitivity shows a peak. So this correspondence indicates the spin correlations are important for this thermal hole effect. Sorry. Later on, thermal hole effect in Kagome, Capilla site has been measured. And here, the thermal hole effect is plotted by this kind of form. So thermal conductivities are normalized by Jaroszynski-Moria interaction and exchange interaction. So it's difficult to see what all curves, experimental curves are collapsed into the single curve. And also, theoretical curves reproduce this temperature dependence, as well as this magnitude. So this indicates that the hole effect is Jaroszynski-Moria plays a role in inducing the final hole effect. And this correspondence indicates that the hole effect arrives from the bosonic spinons in your spin leakage, as explained by this theory. OK, let me summarize my first talk. I showed that the thermal transport measurements are very sensitive and unique probe for detecting itinerant spin excitations. In some spin leakage state, so very highly mobile quasi-particle excitations are observed in two-dimensional triangular lattices. And in two-dimensional Kagome lattices, by measuring a specific thermal hole effect, we observed a unique thermal hole effect due to which can be attributed to the bosonic spinons in your spin leakage. So combining thermal conductivity and thermal hole conductivity, we can make a very strong constraint on the theory for the quantum spin leakage. OK, so in the next lecture, I will talk about another example of the more exotic quasi-particles. So actually, one of them shows the quantized hole effect. Which provides evidence for topological quantum spin leakage state. OK, thank you for your attention.