Zigzag (Ancient Circles Chainsaga Book 1)

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One says to abstain from narcotics and liquors. The other, liberal, translation favored by the great scholar Dr. This is especially true in Japan and China—my goodness, how they throw it down! Now you see, these are—as I say—they are not commandments. They are vows.

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Buddhism has in it no idea of there being a moral law laid down by some kind of cosmic lawgiver. And the reason why these precepts are undertaken is not for a sentimental reason.

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It is that, for anybody interested in the experiments necessary for liberation, these ways of life are expedient. Nothing is more confusing to the mind than taking words too seriously. And finally, to get intoxicated or narcotized—a narcotic is anything like alcohol or opium which makes you sleepy. We come, then, to the final parts of the eightfold path. There is the lifting of the foot. He is self-aware. This is tricky. Ordinary conscious awareness is seeing the world with blinkers on. As we say, you can think only of one thing at a time. Or when you think about the future?

I am aware that I am remembering?

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I am aware that I am thinking about the future? All thoughts are in the present and of the present. And when you discover that, you approach samadhi. Samadhi is the complete state; the fulfilled state of mind. And you will find many, many different ideas among the sects of Buddhists and Hindus as to what samadhi is. Some people call it a trance, some people call it a state of consciousness without anything in it; knowing with no object of knowledge. Some people say that it is the unification of the knower and the known.

All these are varying opinions. I had a friend who was a Zen master, and he used to talk about samadhi, and he said a very fine example of samadhi is a fine horserider. When you watch a good cowboy, he is one being with the horse.

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So an excellent driver in a car makes the car his own body, and he absolutely is with it. So also a fine pair of dancers. They have a way of understanding each other, of moving together, as if they were Siamese twins. Now, when we get to that point in Buddhism, Buddhists do a funny thing, which is going to occupy our attention for a good deal of this seminar. Common or garden atheism is a form of belief, namely that I believe there is no god. The atheist positively denies the existence of any god. All right. A non-god is an inconceivable something or other.

I love the story about a debate in the Houses of Parliament in England—where, as you know, the Church of England is established and, therefore, under the control of the government—and the high ecclesiastics had petitioned Parliament to let them have a new prayerbook. We all believe in some sort of a something somewhere. You have to have an outside, and space outside it, to have a shape. But nonetheless, Jews and Christians persistently make images of God, not necessarily in pictures and statues, but they make images in their minds. And those are much more insidious images.

Do not mistake the finger for the moon. And so Buddha chopped off the finger and undermined all metaphysical beliefs. There are many, many dialogues in the Pali scriptures where people try to corner the Buddha into a metaphysical position. He maintains what is called the noble silence. Sometimes, later, called the thunderous silence—because this silence, this metaphysical silence, is not a void. It is very powerful. This silence is the open window through which you can see not concepts, not ideas, not beliefs, but the very goods.

But if you say what it is that you see, you erect an image and an idol, and you misdirect people. I know it hurts, but it is The Way. That is what cracks the eggshell and lets out the chick. Of course, if you want to stay in the eggshell, you can. This, then, you see, is why Buddhism is in dialogue form: the truth cannot be told. It develops, it grows. As people make all these explorations that the original Buddha suggested, they find out all kinds of new things, they explore the mind, they find out all the tricks of the mind, they—oh, they find out ever so many things, and they begin to teach these things; talk about them.

And nobody can get back to Buddha. You can only go on to Buddha. I mean, occasionally. We ought to be just that simple little acorn. And what it does is this: as it grows—say, it grew from a seed; an acorn—it keeps dropping off new acorns. And that becomes a new seed for another tree. This is fine. Now, let me just warn you: the scholarly study of Buddhism is a magnum opus beyond belief. There are two collections of Buddhist canonical scriptures. We have these collections in Tibetan and Chinese. And then, finally, a few pearls of wisdom are dropped by the Buddha—or else, they sometimes go on for pages, and pages of—actually—very, very subtle and very profound discourse that is not dull if you have a penchant for that kind of thing.

So you see, from time to time, Buddhists get tired of the scriptures. Actually, they keep them in a revolving bookcase in some monasteries. A thing about so high, so wide; it revolves. In Zen monasteries, they have an annual ceremony for reading the scriptures. But they are printed like an accordion. In other words, the pages are connected to each other zig-zag.

Like a slinky moves. And so, each monk is assigned a pile of the volumes—this happens once a year—and they all chant sections of the scripture. But very often, each monk chants a different one. So, you see, Buddhists are funny about scriptures. They respect them, they occasionally read them, but they feel that the writing , the written word, is purely incidental.

It is not the point. And, indeed, it can be a very serious obstacle. You must understand, as one of the fundamental points of Buddhism, the idea of the world as being in flux. Not really. But that whirlpool never, never really holds any water. The water is all the time rushing through it. In the same way, a university—the University of California —what is it?

The students change at least every four years, the faculty changes at a somewhat slower rate, the buildings change—they knock them down and put up new ones—the administration changes. So what is the University of California? A doing of a particular kind. And so in just precisely that way, every one of us is a whirlpool in the tide of existence, and wherein every cell in our body, every molecule, every atom is in constant flux, and nothing can be pinned down.

You know, you can put bands on pigeons, or migrating birds, and identify them and follow them, and find out where they go. Does the light in the refrigerator really go off when we close the door? But this is fundamental, you see, to Buddhistic philosophy. The philosophy of change. From one point of view, change is just too bad. On the one hand, resentment, and on the other, delight. If you resist change—of course, you must to some extent. You have a beautiful girl, and you touch her.

A little bit of resistance, you see, is great. But the human mind, as distinct from most animal minds, is terribly aware of time. And so we think a great deal about the future, and we know that every visible form is going to disappear and be replaced by so-called others. Are these others, others? Or are they the same forms returning? What do you mean by the same? What happens when any great musician plays a certain piece of music? He plays it today, and then he plays it again tomorrow.

Is it the same piece of music, or is it another? And this is as true in our lives as they go on now, from moment to moment, as it would be true of our lives as they appear and reappear again over millions of years. But when you get the lowest audible notes that one can hear on an organ, you feel the shaking. So in the same way as we live now, from day to day, we experience ourselves living at a high rate of vibration, and we appear to be continuous—although there is the rhythm of waking and sleeping.

But the rhythm that runs from generation to generation and from life to life is much slower, and so we notice the gaps.

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So we are living, as it were, on many, many levels of rhythm. This is the nature of change. If you resist it you have dukkha; you have frustration and suffering. It becomes positively beautiful, which is why—in poetry—the theme of the evanescence of the world is beautiful. When Shelley says,. Or is it rather the dome of many-colored glass that shatters? Somehow, you know, the poet has got the intuition. The fact that things are always running out, that things are always disappearing, has some hidden marvel in it. So in the same way, the coming and going of things in the world is marvelous.

They go. Where do they go? They vanish into the mystery. In variance with the dimer systems where the gap is basically isotropic, the properties of DTN strongly depend upon the direction of the magnetic field with respect to the easy magnetization axis. The uniform half-integer spin chain does not present a gap in the triplet excitation spectrum. The fragments of crystal structures are depicted in the insets in each panel. Integer spin chains with sufficiently weak anisotropy are characterized by a nonmagnetic singlet ground state and a nonzero excitation-energy gap.

The interactions of spin, charge and orbital degrees of freedom with the lattice lead to the spin-Peierls transition, charge and orbital order driven transitions. All of them include structural distortion and in every case a loss in elastic energy is compensated by a gain in magnetic energy. The spin-Peierls transition, being the most unusual kind of magnetoelastic transition, relates to the particular quantum mechanical nature of quasi-one-dimensional AFM. Similar to the Peierls transition in quasi-one-dimensional conductors, the spin-Peierls transition integrates spin gap formation and dimerization of the underlying crystal lattice.

Symbols represent the experimental data taken along three principal axes; b The charge ordering transition in NaV 2 O 5 adapted with permission from, ref. The solid lines in b and c represent the Bonner—Fisher curve. The dashed line in panel c represents the Curie—Weiss law. At present, the alternation of exchange interaction within zigzag chain is considered to be responsible for a spin gap. The low temperature crystal structure in NaV 2 O 5 is fixed by both lattice distortion and Coulomb repulsion. A spin-Peierls-like phase transition 61 driven by spin-orbital fluctuations 62 was observed in NaTiSi 2 O 6.

Taking into consideration the orbital degree of freedom, the Hamiltonian of this system can be written as The ground state of this Hamitonian is a dimerized orbital-ordered one hosting the spin singlet on each bond. The condensation of the system in either one of these states explains the appearance of a large singlet-triplet spin gap. The long-range order is the final destination for numerous quasi-one-dimensional magnets not protected by a spin gap.

The ground state of these systems also depends on both signs and values of nearest neighbor J nn and next nearest neighbor couplings J nnn within the chains. Of special interest is LiCuVO 4 which exhibits ferroelectricity at low temperatures and nematicity at high magnetic fields. LiCuVO 4 is an improper ferroelectric with the long-range polar order induced at the onset of a spiral spin order. Finally, for external magnetic fields above H 2 , the helical spin structure is destroyed and the system is paraelectric for all field directions.

The peak of each line is marked by the black triangle. Spin-density wave, spin nematic and saturated field ranges are highlighted by different colors. The dark red area corresponds to an anomalous spin density wave phase; the dark yellow area depicts a nematic phase. The blue line marks the isosbestic field H c1. The brown circles depict the maximum of the spin-nematic correlation function H SN max.

Above H 2 , an incommensurate, collinear spin density wave of bound magnon pairs is stabilized in medium magnetic fields by a FM J nn. In high fields just below the saturation of magnetization, these pairs experience a Bose—Einstein condensation into quantum multipolar states.

One of these states expected just below the saturation H S is a quadrupolar state of magnon pairs called a spin nematic state, analogous to a nematic liquid crystal.

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In a spin nematic state, an energy gap develops in the transverse spin-excitation spectrum making the energy of the two-magnon bound state lower than the energy of the single-magnon state. Three different regions can be identified in these NMR spectra. This phase is characterized by a modulated spin polarization, where the moments are collinear with the external field.

In the field ranges This behavior corresponds to the formation of a homogeneous magnetic state as expected for a spin-nematic state. Excluding well established mechanisms for the spin gap formation, i. According to phase diagram, shown in Fig. The range of possible multipolar nematic phase was narrowed to Isolated magnetic entities consisting of exchange-coupled chains constitute the multitude of spin ladders.

Depending on the ratio of the rung J r and the leg J l exchange interactions, various ground states could be formed in these objects. In the case of the spin-1 ladder the ground state is gapped for any ratio of J r and J l. Of special interest is the Nersesyan—Tsvelik network, which is an extension of the spin-ladder pattern to the layer where both rung J r and plaquette-diagonal J d exchange interaction are taken into account. Both frustration and spatial anisotropy of exchange interactions are essential ingredients of the Nersesyan—Tsvelik model.

The Hamiltonian in this case is. The spatially anisotropic square lattice quantum AFM was analyzed by Starykh and Balents who showed that to realize the Nersesyan—Tsvelik model just the reduction of the coupling of staggered magnetization of different chains is needed, not full elimination. The layered crystal structure of this compound is organized by weakly coupled chains running along the b axis, as shown in Fig.

The interplane exchange interaction along the a axis is considered to be small. The boundary of two-spinon continuum is marked by dashed line adapted with permission from, ref. Another probe of the spin liquid state was Raman spectroscopy which evidenced a gapless continuum of magnetic origin Fig. That same spinon continuum was observed in inelastic neutron scattering Fig. The deviation from this ratio may lead to formation of the Neel state at low temperatures.

Since the inelastic neutron scattering reveals commensurate magnetism along the chains, the order must be incommensurate perpendicular to the chains. Hence, the NO Cu NO 3 3 can be considered as a highly one-dimensional chain compound with frustrated interchain interactions. The introduction of holes into the copper layers leads to frustration of magnetic interactions and formation of resonating dimer singlets, i.

This allowed Anderson advance a concept of high-T C superconductivity in cuprates based on idea of resonating valence bond RVB state. Below T N , the sharp excitations appear at low energies, but the dominant continuum at higher energies remains basically unchanged. It was argued by Kohno, Starykh and Balents, 93 that the sharp excitations represent the spinon bound states, i. The data obtained suggest that Cs 2 CuCl 4 could be placed into close proximity to quantum critical point separating fractional resonating-valence-bond RVB spin liquid and a magnetically ordered state, as shown in Fig.

In a magnetic field, the phase diagram of Cs 2 CuCl 4 has been found to be quite sensitive to smallest interactions. A cascade of energy scales pertinent to Cs 2 CuCl 4 in a magnetic field oriented along the b axis is represented by Fig. Quite a few 2D compounds were considered hosting a quantum spin liquid on a geometrically frustrated kagome lattice. To reveal the intrinsic properties of a kagome layer much better is the local probe, i. The fractional spin excitations in ZnCu 3 OH 6 Cl 2 form flat continuum evidenced in neutron scattering measurements.

This is a signature of a quantum spin liquid. The key issue in this respect is the presence or absence of a spin gap. While it was not established unambiguously, the neutron scattering data set an upper limit for the spin gap value of about 0. The arrow shows the concavity of the M 1 mode adapted with permission from, ref.

The shaded blue region indicates the continuum contribution adapted with permission from, ref. Strong spin-orbit coupling was found to play a key role in the formation of anisotropic bond-dependent interactions on the honeycomb lattice in this case. Despite expectations, however, all these compounds do not have a true spin-liquid ground state because they demonstrate long-range AFM order at low temperatures, preceded by a wide maximum on the temperature dependence of the magnetic susceptibility.

This cannot be related to Kitaev interactions, originating from the direct exchange between the transition metal ions. At the same time, the properties of these compounds at elevated temperatures reflect the proximity to Kitaev model and remain to be of great interest.

Such incoherent spectra were observed in inelastic neutron scattering , Fig. The BKT paradigm formulated initially for the frustrated square lattice can be extended to triangular, kagome and honeycomb systems also. This concept presumes a phase transition from unbound vortex and antivortex state of two-dimensional magnet to the coupled vortex—antivortex phase at low temperatures. Below critical temperature of this transition, the formation of topological defects vortex-antivortex pairs leads to the appearance of additional degree of freedom, i.

The versatile phenomena seen in low-D quantum magnets are just mentioned here in an introductory manner. Each of these phenomena deserves a separate review papers, interested readers are respectfully referred to them. The choice of milestones in the field of low-dimensional magnetism is highly debatable. There cannot be unambiguous criteria for importance, timeliness or impact on the scientific community. Several advanced models and concepts of low-dimensional magnetism, for example the BKT transition or the Kitaev model, are still waiting for a rigorous experimental verification.

Quite recently, a new member of honeycomb iridates family, Cu 2 IrO 3 , becomes available. Although Cu 2 IrO 3 experiences weak magnetic order at 2. The list of chosen spin-gap compounds is given in Table 1. There are not many, and the gapless spin-liquids are even scarcer. Fortunately, every new compound with an exotic ground state and non-trivial excitations brings new colors to the palette of quantum cooperative phenomena in solids and brings new inspiration to researches concentrated on this fascinating topic.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Ising, E. Report on the theory of ferromagnetism. Heisenberg, W. On the theory of ferromagnetism. Bethe, H. Metal theory. Onsager, L. Crystal statistics I. A two-dimensional model with an order-disorder transition. Mermin, N. Absense of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Berezinskii, V. Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group.

Quantum systems. JETP 34 , — Kosterlitz, J. Long range order and metastability in two dimensional solids and superfluids. Solid State Phys. Ordering, metastability and phase transitions in two-dimensional systems. Haldane, F. Continuum dynamics of the 1-D Heisenberg anti-ferromagnet identification with the O 3 non-linear sigma-model. A 93 , — Giamarchi, T. Shastry, B. Exact ground state of quantum-mechanical antiferromagnet.

B , — Miyahara, S. Kageyama, H. Exact dimer ground state and quantized magnetization plateaus in the two-dimensional spin system SrCu 2 BO 3 2. Direct evidence for the localized single-triplet excitations and the dispersive multitriplet excitations in SrCu 2 BO 3 2. Takigawa, M. Matsuda, Y. Corboz, P. Crystals of bound states in the magnetization plateaus of the Shastry — Sutherland model.

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Koga, A. Zayed, M. Matsubara, T. A lattice model of liquid helium, I. Nikuni, T. Bose—Einstein condensation of dilute magnons in TlCuCl 3. Waki, T. Jaime, M. Aczel, A. Samulon, E. Manaka, H. Field-induced magnetic long-range order in the ferromagnetic-antiferromagnetic alternating Heisenberg chain system CH 3 2 CHNH 3 CuCl 3 observed by specific heat measurements.

Zapf, V. Bose—Einstein condensation in quantum magnets. Ordered magnetic phases of the frustrated spin-dimer compound Ba 3 Mn 2 O 8. B 77 , Lieb, E. Two soluble models of an antiferromagnetic chain. Bonner, J. Linear magnetic chains with anisotropic coupling. A , — Belik, A. Solid State Chem. Johannes, M. Sr 2 Cu PO 4 2 : a real material realization of the one-dimensional nearest neighbor Heisenberg chain.

B 74 , Excitation spectra of the linear alternating antiferromagnet. B 25 , — Johnston, D. B 61 , — Kokado, S. Jpn 68 , — Borras-Almenar et al. Alternating chains with ferromagnetic and antiferromagnetic interactions. Theory Magn. He, Z. BaCu 2 V 2 O 8 : Quasi-one-dimensional alternating chain compound with a large spin gap. B 69 , Ghoshray, K. B 71 , Koo, H. Salunke, S.

Klyushina, E. B 93 , Nonlinear theory of large-spin Heisenberg antiferromagnets: semiclassically quantized solitons of the one-dimensional easy-axis Neel state. Botet, R. Ground state properties of a spin-1 antiferromagnetic chain. B 27 , — Nightingale, M. Gap of the linear spin-1 Heisenberg antiferromagnet: a Monte Carlo calculation. B 33 , — Uchiyama, Y. Spin-vacancy-induced long-range order in a new Haldane-gap antiferromagnet. Zheludev, A. Magnetic excitations in coupled Haldane spin chains near the quantum critical point. B 62 , — Bray, J. Observation of a spin-Peierls transition in a Heisenberg antiferromagnetic linear-chain system.

Hase, M. Bulaevskii, L. Spin — Peierls transition in magnetic field. Solid State Commun. Nishi, M. Neutron-scattering study on the spin-Peierls transition in a quasi-one-dimensional magnet CuGeO 3. B 50 , — Northby, J. Field-dependent differential susceptibility studies on tetrathiafulvalene-AuS 4 C 4 CF 3 4 : Universal aspects of the spin-Peierls phase diagram. Magnetic phase diagram of the spin-Peierls cuprate CuGeO 3.

B 48 , — Nojiri, H. Observation of magnetization saturation of CuGeO 3 in ultrahigh magnetic fields up to T. B 55 , — Isobe, M. Ohama, T. B 59 , — Sawa, H. X-ray anomalous scattering study of a charge-ordered state in NaV 2 O 5. Ohwada, K. Redhammer, G. Acta Cryst. Konstantinovic, M. Silverstein, H.